# Long-Term Load Forecasting Based on a Time-Variant Ratio Multiobjective Optimization Fuzzy Time Series Model

Authors: Xiaojuan Liu; Jian’an Fang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Load forecasting problem is a complex nonlinear problem linked with economic and weather factors. Long-term load forecasting provides useful information for maintenance scheduling, adequacy assessment, and limited energy resources for electrical power systems. Fuzzy time series forecasting models can be used for long-term load forecasting. However, the interval length has been chosen arbitrarily in the implementations of known fuzzy time series forecasting models, which has an important impact on the performance of these models. In this paper, a time-variant ratio multiobjective optimization fuzzy time series model (TV-RMOP) is proposed, and its performance is tested on the prediction of enrollment at the University of Alabama. Results clearly promote the forecasting accuracy as compared to the conventional models. A genetic algorithm is used to search for the length of intervals based on the training data while Pareto optimality theory provides the necessary conditions to identify an optimal one. The TV-RMOP model is applied for the long-term load forecasting in Shanghai of China.


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## Body

## 1. Introduction

Electric power load forecast has been a research topic for many decades and the accuracy of load forecast is crucial to electricity power industry due to its direct influence on generating planning. During the past years, there have been numerous attempts to improve the accuracy of load forecasting methods [1, 2]. The forecasting model and methods can be divided into two categories: statistical methods and computational intelligence algorithms. The former method include regression analysis and time series [3, 4]. The latter include neural network, expert system, fuzzy logical, and so forth [5–7]. Now more and more scholar and researchers pay much attention to uncertainty hybrid method [8–10].There are three different types of electric load forecasting depending on the time horizon and the operating decision that needs to be made: short-, medium-, and long-term forecasting. In general, long term forecasting includes prediction making for a range more than a year and is needed for power supply and delivery system. Therefore, the weather factor is not the main factor to affect the long-term load forecast, only using historical load data can predict the 1 year ahead load. Fuzzy time series [11, 12] prediction method is likely to get a satisfactory result.Fuzzy time series has been widely studied for recent years for the aim of forecasting. The basic process of conventional model is shown in Figure1(a). Related studies mainly focus on three factors: the partition of discourse, the content of forecasting rules, and the methods of defuzzification, all of which influence the prediction accuracy of forecasting models. A number of fuzzy time series forecasting models have been presented since Song and Chissom [13, 14] first presented the concepts of fuzzy time series based on the fuzzy set theory. However, the length of intervals has been chosen arbitrarily in known models. This paper focuses on the partition of discourse in TV-RMOP. We propose a novel approach to determine the length of intervals based on multiobjective optimization method in order to obtain more accurate forecasts in fuzzy time series. The proposed approach is based on two-parameter constrained multiobjective optimization. A genetic algorithm is used to search for the two parameters based on the training data. The TV-RMOP model is shown in Figure 1(b). Its performance is tested on the prediction of enrollment at the University of Alabama and applied for the long-term load forecasting in Shanghai of China. Results investigated that TV-RMOP model considerably improves the forecasting performance.Two distinct prediction processes of fuzzy time series forecasting.
(a)
Conventional process
(b)
Our processThe rest of this paper is organized as follows. Section2 introduces the methodology background briefly. The presented model is described, and the forecasting results on the prediction of enrollment at the University of Alabama are presented and discussed in Section 3. In Section 4, TV-RMOP model is applied for the long-term load forecasting in Shanghai of china. Finally, conclusions are presented in Section 5.


## 2. Background

### 2.1. Fussy Time Series

The concepts of fuzzy time series are described as follows [13, 14].LetU be the universe of discourse, where U={u1,u2,…,un}. A fuzzy set A of U is defined as A=fA(u1)/u1+fA(u2)/u2+⋯+fA(un)/un, where fA is the membership function of the fuzzy set A, fA:U→[0,1]. ui is a generic elements of fuzzy set A. fA(ui) is the degree of belongingness of  ui to A, fA(ui)∈[0,1], and 1≤i≤n.Definition 1.
LetY(t)(t=…,0,1,2,…) be the universe of discourse and also a subset of real numbers. It is assumed that fi(t) is defined on Y(t) and F(t) is the collection of fi(t), then F(t) is called a fuzzy time series on Y(t).Definition 2.
It is assumed thatF(t) is a fuzzy time series and F(t)=F(t-1)×R(t,t-1), where R(t,t-1) is a fuzzy relation and × is an operator which is caused by F(t-1). The relationship between F(t) and F(t-1) can be denoted by F(t-1)→F(t) when F(t)=F(t-1)×R(t,t-1) is the first-order fuzzy time series model of  F(t).Definition 3.
LetF(t) be a fuzzy time series. For any t, F(t)=F(t-1) and F(t) have only finite elements and therefore F(t) is a time-invariant fuzzy time series; otherwise, it is a time-variant fuzzy time series.


### 2.2. Multiobjective Optimization and Pareto Points

It is common in decision-making problems to have a set of parameters that play a dominant role in the process. It is also desirable for the decision made to be the best possible with respect to the available data. In the context of mathematics, the best decision coincides with the optimization of a function of the decision parameters, known as cost function or objective.On the one hand, if the goal is to optimize a single cost function, then the problem is classified as single objective optimization. On the other hand, the process of simultaneously optimizing two or more conflicting objectives, subject to certain constraints, is identified as multiobjective optimization [17–19].Such problems are called multiobjective optimization problems and formulated as follows:(1)minxF(x)=[f1(x),f2(x),fk(x)]subjecttohi(x)≤0,i=0,1,…,n1gl(x)=0,l=0,1,…,n2,
where k denotes the number of objective functions; n1, n2 identify the number of inequality and equality constraints, respectively. x is an n-dimensional vector in the solution space Xn, and x=(x1,x2,…,xn)∈Xn.Multiobjective optimization problems do not have a single global solution in contrast to the single objective problems. As a result, it is required to specify a number of criteria-conditions for which a set of solutions can be identified as optimal. In general, the optimal solutions forF(x) can be obtained based on the Pareto optimality theory. The set of optimal solutions, which is known as the Pareto optimal set X*, are those solutions such that each of these cannot improve any objective function in F(x) without the worsening of some other objective functions at the same time. The optimal solution set is defined as
(2)X*={x*∈X∣doesnotexistx∈X,F(X)≤F(X*)}.
The corresponding points, which are known as the nondominated points, form the Pareto front in the objective space. Since there exists different tradeoff solutions to minimize F(x), one possible way to solve the multiobjective problem is to select one solution manually from the Pareto optimal set.


## 2.1. Fussy Time Series

The concepts of fuzzy time series are described as follows [13, 14].LetU be the universe of discourse, where U={u1,u2,…,un}. A fuzzy set A of U is defined as A=fA(u1)/u1+fA(u2)/u2+⋯+fA(un)/un, where fA is the membership function of the fuzzy set A, fA:U→[0,1]. ui is a generic elements of fuzzy set A. fA(ui) is the degree of belongingness of  ui to A, fA(ui)∈[0,1], and 1≤i≤n.Definition 1.
LetY(t)(t=…,0,1,2,…) be the universe of discourse and also a subset of real numbers. It is assumed that fi(t) is defined on Y(t) and F(t) is the collection of fi(t), then F(t) is called a fuzzy time series on Y(t).Definition 2.
It is assumed thatF(t) is a fuzzy time series and F(t)=F(t-1)×R(t,t-1), where R(t,t-1) is a fuzzy relation and × is an operator which is caused by F(t-1). The relationship between F(t) and F(t-1) can be denoted by F(t-1)→F(t) when F(t)=F(t-1)×R(t,t-1) is the first-order fuzzy time series model of  F(t).Definition 3.
LetF(t) be a fuzzy time series. For any t, F(t)=F(t-1) and F(t) have only finite elements and therefore F(t) is a time-invariant fuzzy time series; otherwise, it is a time-variant fuzzy time series.


## 2.2. Multiobjective Optimization and Pareto Points

It is common in decision-making problems to have a set of parameters that play a dominant role in the process. It is also desirable for the decision made to be the best possible with respect to the available data. In the context of mathematics, the best decision coincides with the optimization of a function of the decision parameters, known as cost function or objective.On the one hand, if the goal is to optimize a single cost function, then the problem is classified as single objective optimization. On the other hand, the process of simultaneously optimizing two or more conflicting objectives, subject to certain constraints, is identified as multiobjective optimization [17–19].Such problems are called multiobjective optimization problems and formulated as follows:(1)minxF(x)=[f1(x),f2(x),fk(x)]subjecttohi(x)≤0,i=0,1,…,n1gl(x)=0,l=0,1,…,n2,
where k denotes the number of objective functions; n1, n2 identify the number of inequality and equality constraints, respectively. x is an n-dimensional vector in the solution space Xn, and x=(x1,x2,…,xn)∈Xn.Multiobjective optimization problems do not have a single global solution in contrast to the single objective problems. As a result, it is required to specify a number of criteria-conditions for which a set of solutions can be identified as optimal. In general, the optimal solutions forF(x) can be obtained based on the Pareto optimality theory. The set of optimal solutions, which is known as the Pareto optimal set X*, are those solutions such that each of these cannot improve any objective function in F(x) without the worsening of some other objective functions at the same time. The optimal solution set is defined as
(2)X*={x*∈X∣doesnotexistx∈X,F(X)≤F(X*)}.
The corresponding points, which are known as the nondominated points, form the Pareto front in the objective space. Since there exists different tradeoff solutions to minimize F(x), one possible way to solve the multiobjective problem is to select one solution manually from the Pareto optimal set.


## 3. Modeling and Verification

### 3.1. Proposed TV-RMOP Model

Time series is divided into two parts. One part is used for training purpose while the other part is used for testing. The proposed model was applied to the training data to make estimation and then the predicted values were found for the test data. In the training phase, five measures are adopted to evaluate the model prediction accuracy with the intention to formulate the multiobjective problem. These measures are objective functions with parameters and defined as follows:(3)RootMeanSquareError(RMSE):RMSE(·)=1N∑t=1N(Rt-Pt)2MeanAbsoluteError(MAE):MAE(·)=1N∑t=1N|Rt-Pt|MeanAbsolutePercentageError(MAPE):MAPE(·)=100N∑t=1N|Rt-PtRt|MaximumAbsolutePercentageError(MAP):MAP(·)=maxt=1,…,N(100×|Rt-PtRt|)TheilInequalityCoefficient(THEIL):THEIL(·)=RMSE(∑t=1N(Rt)2)/N,
where  Rt is the actual value and Pt is the predicted value at time instant t. N is the number of predictions made for a prespecified time interval. All measures should be minimized in the optimization procedure.As shown in Figure2, the TV-RMOP model can be summarized as follows.Figure 2
Proposed time-variant ratio multiobjective optimization fuzzy time series model.Step 1.
Define the universe of discourse and intervals using equations defined in the upper-right dashed box. Whereinitial  and ratio  are two unknown parameters.Step 2.
Define fuzzy sets based on the universe of discourse and fuzzify the historical data.Step 3.
Establish fuzzy relationships:Ai→Aj.Step 4.
Forecast. LetF(t-1)=Ai; the forecast is the midpoint of the interval which corresponds to Ai.Step 5.
Calculate RMSE, MAE, MAPE, MAP, and THEIL, respectively. Each of them is a function with variableinitial and ratio.In the training phase, the training samples (Dtrain) are used to calculate the above five functions. The determination of variable initial and ratio is formulated as a multiobjective optimization problem:
(4)minF(initial,ratio)=[RMSE,MAE,MAPE,MAP,THEIL]subjecttoinitial∈(a,b),ratio∈(A,B).In the test phase, an optimal solution is brought into the model, and the forecast value is determined in the same way as the training phase.Genetic algorithm can be used to search for a solution of multiobjective optimization problems. The process is shown in the dashed box on the left in Figure2. In this paper we use a MATLAB function “gamultiobj” to optimize the initial and ratio parameters.


### 3.2. Verification of Proposed Model

In order to show what we have achieved from the TV-RMOP model, we apply it to the enrollment of University of Alabama and then have compared the obtained results with those obtained from Chen’s method [20]. Table 1 lists the enrollment from 1971 to 1992. The observations from 1971 to 1989 are used for the training (Dtrain), while the observations from 1990 to 1992 are used for test.Table 1
The enrollment data.
Year
Enrollments
1971
13055
1972
13563
1973
13867
1974
14696
1975
15460
1976
15311
1977
15603
1978
15861
1979
16807
1980
16919
1981
16388
1982
15433
1983
15497
1984
15145
1985
15163
1986
15984
1987
16859
1988
18150
1989
18970
1990
19328
1991
19337
1992
18876The minimal value ofDtrain is 13055, then [12000,13000] is taken as constrains of variable initial. (0,0.2] is taken as constrains of the other variable ratio, because when the ratio is greater than 0.2, the length of interval will be too large, and  the number of intervals will decrease. It results in diminishing the fluctuation in fuzzy time series. Then the problem can be formulated as
(5)minF(initial,ratio)=[RMSE,MAE,MAPE,MAP,THEIL]subjecttoinitial∈[12000,13000],ratio∈(0,0.2].We use a MATLAB function “gamultiobj” to solve (5). The initialization of genetic algorithm parameters is set as follows. The fraction of population on non-dominated front is 0.1, the number of individuals is 50, maximum number of generations allowed is 100, and termination tolerance on fitness function value is 1e-100. The Pareto solutions of training phase are provided in Table 2 (seen in Figure 3).Table 2
Pareto solutions of enrollment.
Initial
Ratio
RMSE
MAE
MAPE
MAP
THEIL
12266.7
0.0621
571.9
484.4
2.9745
5.8418
0.0359
12230.4
0.0530
573.9
486.0
2.9983
5.6954
0.0360
12135.5
0.0455
532.7
451.8
2.8276
6.6546
0.0334
12846.0
0.0676
568.0
501.2
3.1217
5.2926
0.0357
12266.7
0.0620
551.5
459.0
2.8142
5.8748
0.0346Figure 3
Pareto front of TV-RMOP model for enrollment.We select 12135.5 and 0.0455 as optimal parameters to forecast the enrollment from 1990 to 1992. For this initial and ratio, the universe of discourse is defined as[12135.5,19694.1]. The intervals are finally defined as: u1=[12135.5,12681.6], u2=[12681.6,13252.3],   u3=[13252.3,13848.6], u4=[13848.6,14471.8],    u5=[14471.8,15123], u6=[15123,15803.6], u7=[15803.6,16514.7], u8=[16514.7,17257.9], u9=[17257.9,18034.5], u10=[18034.5,18846], u11=[18846,19694.1]. Tables 3 and 4 list the forecast results and corresponding measures, respectively. The best RMSE value is 500 with the interval length of 500 for Chen’s method [20]. The RMSE value is 425.46 for Kuo’s model [15] and 327.54 for Wong’s model [16]. The proposed model produces 233.5 for RMSE. Obviously, the TV-RMOP model gives more accurate forecast than [15, 16, 20].Table 3
The forecasting results forDtest.
Year
Actual enrollment
[15]
[16]
TV-RMOP model
1990
19328
18685
18970
19271
1991
19337
19138
19306
19271
1992
18876
19176
19315
19271Table 4
The measures forDtest.
Year
Actual enrollment
[15]
Initial
Ratio
RMSE
MAE
MAPE
MAP
THEIL
12135.5
0.0455
233.5
172.7
0.9098
2.79
0.01212


## 3.1. Proposed TV-RMOP Model

Time series is divided into two parts. One part is used for training purpose while the other part is used for testing. The proposed model was applied to the training data to make estimation and then the predicted values were found for the test data. In the training phase, five measures are adopted to evaluate the model prediction accuracy with the intention to formulate the multiobjective problem. These measures are objective functions with parameters and defined as follows:(3)RootMeanSquareError(RMSE):RMSE(·)=1N∑t=1N(Rt-Pt)2MeanAbsoluteError(MAE):MAE(·)=1N∑t=1N|Rt-Pt|MeanAbsolutePercentageError(MAPE):MAPE(·)=100N∑t=1N|Rt-PtRt|MaximumAbsolutePercentageError(MAP):MAP(·)=maxt=1,…,N(100×|Rt-PtRt|)TheilInequalityCoefficient(THEIL):THEIL(·)=RMSE(∑t=1N(Rt)2)/N,
where  Rt is the actual value and Pt is the predicted value at time instant t. N is the number of predictions made for a prespecified time interval. All measures should be minimized in the optimization procedure.As shown in Figure2, the TV-RMOP model can be summarized as follows.Figure 2
Proposed time-variant ratio multiobjective optimization fuzzy time series model.Step 1.
Define the universe of discourse and intervals using equations defined in the upper-right dashed box. Whereinitial  and ratio  are two unknown parameters.Step 2.
Define fuzzy sets based on the universe of discourse and fuzzify the historical data.Step 3.
Establish fuzzy relationships:Ai→Aj.Step 4.
Forecast. LetF(t-1)=Ai; the forecast is the midpoint of the interval which corresponds to Ai.Step 5.
Calculate RMSE, MAE, MAPE, MAP, and THEIL, respectively. Each of them is a function with variableinitial and ratio.In the training phase, the training samples (Dtrain) are used to calculate the above five functions. The determination of variable initial and ratio is formulated as a multiobjective optimization problem:
(4)minF(initial,ratio)=[RMSE,MAE,MAPE,MAP,THEIL]subjecttoinitial∈(a,b),ratio∈(A,B).In the test phase, an optimal solution is brought into the model, and the forecast value is determined in the same way as the training phase.Genetic algorithm can be used to search for a solution of multiobjective optimization problems. The process is shown in the dashed box on the left in Figure2. In this paper we use a MATLAB function “gamultiobj” to optimize the initial and ratio parameters.


## 3.2. Verification of Proposed Model

In order to show what we have achieved from the TV-RMOP model, we apply it to the enrollment of University of Alabama and then have compared the obtained results with those obtained from Chen’s method [20]. Table 1 lists the enrollment from 1971 to 1992. The observations from 1971 to 1989 are used for the training (Dtrain), while the observations from 1990 to 1992 are used for test.Table 1
The enrollment data.
Year
Enrollments
1971
13055
1972
13563
1973
13867
1974
14696
1975
15460
1976
15311
1977
15603
1978
15861
1979
16807
1980
16919
1981
16388
1982
15433
1983
15497
1984
15145
1985
15163
1986
15984
1987
16859
1988
18150
1989
18970
1990
19328
1991
19337
1992
18876The minimal value ofDtrain is 13055, then [12000,13000] is taken as constrains of variable initial. (0,0.2] is taken as constrains of the other variable ratio, because when the ratio is greater than 0.2, the length of interval will be too large, and  the number of intervals will decrease. It results in diminishing the fluctuation in fuzzy time series. Then the problem can be formulated as
(5)minF(initial,ratio)=[RMSE,MAE,MAPE,MAP,THEIL]subjecttoinitial∈[12000,13000],ratio∈(0,0.2].We use a MATLAB function “gamultiobj” to solve (5). The initialization of genetic algorithm parameters is set as follows. The fraction of population on non-dominated front is 0.1, the number of individuals is 50, maximum number of generations allowed is 100, and termination tolerance on fitness function value is 1e-100. The Pareto solutions of training phase are provided in Table 2 (seen in Figure 3).Table 2
Pareto solutions of enrollment.
Initial
Ratio
RMSE
MAE
MAPE
MAP
THEIL
12266.7
0.0621
571.9
484.4
2.9745
5.8418
0.0359
12230.4
0.0530
573.9
486.0
2.9983
5.6954
0.0360
12135.5
0.0455
532.7
451.8
2.8276
6.6546
0.0334
12846.0
0.0676
568.0
501.2
3.1217
5.2926
0.0357
12266.7
0.0620
551.5
459.0
2.8142
5.8748
0.0346Figure 3
Pareto front of TV-RMOP model for enrollment.We select 12135.5 and 0.0455 as optimal parameters to forecast the enrollment from 1990 to 1992. For this initial and ratio, the universe of discourse is defined as[12135.5,19694.1]. The intervals are finally defined as: u1=[12135.5,12681.6], u2=[12681.6,13252.3],   u3=[13252.3,13848.6], u4=[13848.6,14471.8],    u5=[14471.8,15123], u6=[15123,15803.6], u7=[15803.6,16514.7], u8=[16514.7,17257.9], u9=[17257.9,18034.5], u10=[18034.5,18846], u11=[18846,19694.1]. Tables 3 and 4 list the forecast results and corresponding measures, respectively. The best RMSE value is 500 with the interval length of 500 for Chen’s method [20]. The RMSE value is 425.46 for Kuo’s model [15] and 327.54 for Wong’s model [16]. The proposed model produces 233.5 for RMSE. Obviously, the TV-RMOP model gives more accurate forecast than [15, 16, 20].Table 3
The forecasting results forDtest.
Year
Actual enrollment
[15]
[16]
TV-RMOP model
1990
19328
18685
18970
19271
1991
19337
19138
19306
19271
1992
18876
19176
19315
19271Table 4
The measures forDtest.
Year
Actual enrollment
[15]
Initial
Ratio
RMSE
MAE
MAPE
MAP
THEIL
12135.5
0.0455
233.5
172.7
0.9098
2.79
0.01212


## 4. Applying TV-RMOP Model on Long Term Load Forecasting

Long term load forecasting is needed for power supply and delivery system.  We use twenty years (1990~2010) load of Shanghai region in China for forecast (data from the 2011 Shanghai statistical yearbook). The load is shown in Table5. Where loads from 1990 to 2007 are used for the training (Dtrain), and loads from 2008 to 2010 are used for test (Dtest).Table 5
Load in Shanghai region.
Year
Load (hundred million Kwh)
1990
264.74
1991
288.78
1992
317.38
1993
345.86
1994
337.3
1995
403.27
1996
430.4
1997
454.26
1998
482.94
1999
501.2
2000
559.42
2001
592.99
2002
645.71
2003
745.97
2004
821.44
2005
921.97
2006
990.15
2007
1072.38
2008
1138.22
2009
1153.38
2010
1295.87The minimal value ofDtrain is 264.74, [240, 260] is taken as constrains of variable initial. (0, 0.4] is taken as constrains of the other variable ratio. Then the problem can be formulated as
(6)minF(initial,ratio)=[RMSE,MAE,MAPE,MAP,THEIL]subjecttoinitial∈[240,260],ratio∈(0,0.4].The initialization of genetic algorithm parameters is set as follows. The fraction of population on non-dominated front is 0.1, the number of individuals is 50, maximum number of generations allowed is 100, and termination tolerance on fitness function value is1e-100. The Pareto solutions of training phase are provided in Table 6, also seen in Figure 4.Table 6
Pareto solutions of load.
Initial
Ratio
RMSE
MAE
MAPE
MAP
THEIL
257.68
0.2317
45.28
34.47
5.8185
12.184
0.0719
250.80
0.2400
44.06
34.49
5.9488
13.627
0.0699
256.51
0.2329
45.22
34.54
5.8476
12.448
0.0718
250.80
0.2400
44.06
34.49
5.9488
13.627
0.0699
252.48
0.2380
44.24
34.30
5.8942
13.273
0.0702Figure 4
Pareto front of TV-RMOP model for load forecasting.Select 250.8 and 0.24 as optimal parameters to forecast the loads from 2008 to 2010. Tables7 and 8 list the forecast results and corresponding measures, respectively. It is seen that the MAPE and MAP measures are 7.45% and 10.29%, which meet the actual demand.Table 7
The forecasting results forDtest.
Year
Actual load
TV-RMOP model
2008
1138.22
1021.1
2009
1153.38
1266.2
2010
1295.87
1266.2Table 8
The measures forDtest.
Initial
Ratio
RMSE
MAE
MAPE
MAP
THEIL
250.8
0.24
95.43
86.53
7.45
10.29
0.07966


## 5. Conclusions

Fuzzy time series forecasting method are getting quite popular in recent years. Most studies focus on the forecasting rules and methods of defuzzification. How to determine the lengths of intervals is less studied. The decision on what the lengths will be is important for forecasting accuracy.In this study, a new model TV-RMOP is proposed, which formulates the forecasting as a multiobjective optimization problem and solve it by the use of genetic algorithms. The optimal lengths of intervals can be determined in the proposed model. The model is tested for forecasting the enrollment of University of Alabama. The experimental results show that the TV-RMOP model is more accurate than existing models. Finally, we apply this model for long-term load forecasting of Shanghai Region with satisfactory result.



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