Forecasting Annual Power Generation Using a Harmony Search Algorithm-Based Joint Parameters Optimization Combination Model

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Introduction
The electric power industry is the basic industry for both national economy and social development.
Electrification is an important index for assessing a country's level of modernization.The rapid development of the power industry means the rapid growth of installed capacity and generation capacity.Power generation forecasting plays an important role in national and international electric power planning, which also provides the basis of decision making for the government and the power industry development plan.
First, due to the increase in continuous sustainable positive economic growth rate and large scale industrialization, worldwide electricity consumption is quickly rising [1].In order to meet growing electricity demand, more accurate power generation forecasting is needed for future power planning.
Second, the power generation sector, mainly based on fossil-fueled generation forms, is a typical high energy consumption section.The accelerating economic development leads to increasing energy demand for power generation which results in a series of adverse effects such as air pollution and greenhouse gas (GHG) emission [2].A particularly large fraction of CO 2 emissions, the most important anthropogenic GHG, comes from combustion of fossil fuels at power plants [3].The contribution of power generation systems to global energy-related CO 2 emissions increased from 32.67% (7.41 Gt CO 2 ) in 1997 to 41% (11.9 Gt CO 2 ) in 2007 [4].The effect of power generation on climate change has become a key current issue for researchers and policymakers, so for the national energy conservation and environmental protection, accurate power generation is also required.
Third, to respond to global climate change and GHG emissions, measures to realize low carbon electric power sector have been taken, including fuel switching, improving energy efficiency, renewable energy development and deployment and demand side management (DSM) programs, etc. [5].On the power generation side, many countries have enacted decrees to raise the renewable energy generation share in their power generation systems.The characteristics of renewable energy sources, such as unstability and intermittence cause many difficulties for power generation forecasting.It is meaningful and challenging to obtain more accurate power generation predictions under the circumstance of mixed existence of traditional generation forms and various renewable energy generation forms.
In the last few decades, abundant literature [6][7][8] has focused on power generation forecasting using different classical methods so as to avoid electricity shortages and guarantee adequate infrastructures.The major shortcoming of traditional methods such as regression and time series is their limited accuracy, partially resulting from the use of linear model structures or the predominant use of static nonlinear function relationships.Due to the development of artificial intelligence techniques, artificial neural network (ANN) forecasting models and ANNs combining wavelet, optimization and fuzzy techniques are developed for power generation forecasting [9][10][11][12][13][14].The ANN technique, which is inspired on the biological neural system, represents higher nonlinearity between independent and dependent variables [15].The ANN models can treat nonlinear issues with capability to learn, store and recall information based on a given training dataset [16].However, the accuracy of ANN models is limited because the forecasting accuracy depends on the scale of the training data sets and the inadequacy of these data sets will reflect over the entire problem.Moreover, the hidden layers in ANNs are difficult to explain and they easily achieve local optimal solutions due to the random selection of initial weights [17].
No single forecasting method has been found to outperform other models in all situations since each single model with its own particular advantages and disadvantages cannot identify the true process exactly [18].The purpose of combining forecasts from different models is that this can synthesize the information of each individual forecast into a composite one, which is often regarded as a successful alternative to just using an individual method [19].The combination technique was pioneered by Bates and Granger [20], and applications of combination forecasting can be found in many fields.It is less risky in practice to combine forecasts than to select an individual forecasting method.Moreover, it is proved that the combination forecasting model outperforms the poorest individual forecast, and sometimes even performs better than the best individual model [21].
In electric power systems, power generation time series are rarely pure linear or nonlinear, as they often contain both linear and nonlinear patterns, so no single model is best to treat these uncertain data sequences.That is the main purpose to propose a power generation combination forecasting model.In the existing combination forecasting field, much of the literature has focused on how to determine the combination forecasting weights.The common combination weights determination methods include simple average combination, variance covariance combination, Granger and Ramanthan regression method, and the Discounted Mean Square Forecast Error (DMSFE) combination.No researcher has yet paid attention to the form of a single model in combination forecasting methods, i.e., the single forecast model often adopts a fixed form.In other words, the combination forecasting weights and the form of the single forecasting model are not combined to adjust and adapt to different forecasting issues.In this paper, we proposed a novel Harmony Search (HS) algorithm-based joint parameters optimization combination model.The motivation of the combination model comes from the following aspects: first, the single forecasting model adopts a power function form instead of the traditional fixed form, and the exponential parameter in power functions can be adjusted under certain criteria.Second, the exponential parameter and the combination weights, called joint parameters, are adjusted simultaneously.Through adjusting these joint parameters, the combination forecasting model can reach the best results.Third, the optimal values of joint parameters are determined by using the HS algorithm.
The Harmony Search (HS) algorithm, as a recently emerging metaheuristic technique mimicking the improvisation behavior of musicians [22], is considered a novel successful evolutionary algorithm.The HS algorithm has been successfully applied to many optimization problems in the computation and engineering fields [23][24][25].One of key successful factors of the algorithm is the use of a novel stochastic derivative which can be used even for discrete variables.Instead of a traditional calculus-based gradient, the HS algorithm utilizes a musician's experience as a derivative in searching for an optimal solution.The advantages of the HS algorithm are that it may escape local optima and overcome the drawback of GA's building block theory which works well only if the relationship among variables in a chromosome is carefully considered.Therefore, this paper attempts to use a HS algorithm to optimize the joint parameters in a combination forecasting model in order to improve the forecasting accuracy.Cases are then employed to test the performance of the proposed model.The rest of the paper is organized as follows: Section 2 introduces the joint parameters optimization combination model, Harmony Search algorithm and the HS based joint parameters optimization combination model.The empirical simulation and results analysis are presented in Section 3. Finally, Section 4 gives our conclusions.

Joint Parameters Optimization Combination Model (JPOC)
From the point of view of system identification and modeling, the objective of modeling of a certain system is to determine a model similar to the measured system from a given set of model classes on the basis of the input and output data [26].In other words, the task of system modeling is to find a model which can describe the system characteristics and fit future development trends as accurately as possible.For a practical forecasting issue, it is not easy to exactly identify the future trends of the time series sequence, so a single forecast model cannot always fit the series data better for all situations [18].
Inspired by the system identification and modeling theory, a nonlinear combination model is proposed in our work to forecast the power generation sequence.Adopting a nonlinear model to describe the power generation forecasting model is more appropriate than a linear one since in essence the power generation growth trend is nonlinear.In the nonlinear combination model, the single model adopts a power function form.It is very hard to solve a nonlinear model using the traditional analytical methods.The process of finding the coefficients and exponents of the nonlinear model could be regarded as an optimization problem.Artificial intelligence methods provide an effective approach to solve such optimization problems.A novel intelligence optimization method-Harmony Search algorithm-is introduced to access the optimal exponential parameters of the single power function model and the combination forecasting weights simultaneously.
In this section, the joint parameters optimization combination model is described.The joint parameters optimization combination model includes single model parameter optimization and combination weight optimization.
The form of joint parameters optimization combination model is written as follows: where t y  denotes the combined forecasting value for the time period t, ) ( ˆi t y is the ith forecasting value for the same period, k is the number of forecasts to be combined, ù i is the combination forecasting weight assigned to the ith participating model, n i is the exponent of the ith single model.The optimal value of ù i and n i can be determined by the Harmony Search algorithm optimization technique.In all, the advantages of the proposed HS algorithm-based joint parameters optimization combination model are as follows: first, it is presented based on nonlinear theory which reflects the nonlinear essence of the power generation sequence.Second, the joint optimal parameters, including exponent coefficient and combination weights, could only be determined simultaneously through artificial intelligence techniques and cannot be solved through traditional analytical methods.Third, the HS algorithm imitates the musical improvisation process in which seeking a perfect state of harmony between different instruments according to aesthetic standard is analogous to seeking a global optimum between different variables according to an objective function in optimization techniques.This means the HS algorithm is easily understood compared with other optimization algorithms.

Harmony Search (HS) Algorithm
The Harmony Search (HS) algorithm, proposed by Geem et al., is a phenomenon-mimicking algorithm inspired by the improvisation process of musicians [22].Compared with other heuristic optimization algorithms, it behaves with excellent effectiveness and robustness and presents lots of advantages when applied to optimization problems [27,28].Scheme 1 shows the HS algorithm optimization procedures consisting of Steps 1-5.

Scheme 1. Harmony Search (HS) optimization procedures.
Step 1. Initialize the optimization problem and algorithm parameters: where f(x) is the objective function; x is the set of each design variable (x i ); X i is the set of the possible range of values for each design variable; N is the number of design variables.In addition, the HS algorithm parameters including harmony memory size (HMS), harmony memory considering rate (HMCR), pitch adjusting rate (PAR), the lower bounds (lb) and upper bounds (ub) for each decision variable and termination criterion should also be specified in this step.
The HM is a location storing all the solution vectors.In this step, the HM matrix is filled with randomly generated solution vectors and sorted by the values of the objective function f(x).
Step 3. Improvise a new harmony from the HM.
A new harmony vector is generated based on three rules: memory consideration, pitch adjustment and random selection.
Step 4. Update the HM.On condition that the new harmony vector showed better fitness function than the worst harmony in the HM, the new harmony is included in the HM and the existing worst harmony is excluded from the HM.
Step 5. Repeat steps 3 and 4 until the termination criterion is satisfied.

HS Based Joint Parameters Optimization Combination Model (HS Based JPOC Model)
The HS-based joint parameters optimization combination model (HS-based JPOC model) is described in this section.The optimization objective function is specified as the mean absolute percentage error (MAPE).The MAPE is measure of accuracy in a fitted time series value in statistics, specifically trending.It usually expresses accuracy as a percentage, eliminating the interaction between negative and positive values by taking absolute operation [29], shown in Equation ( 2): where t y is the actual value for tth period; t y  represents its forecasting result which can be calculated through Equation ( 1); and T is the number of data used for the MAPE calculation.Then the optimization objective function is expressed as follows: The optimal values of the joint parameters ù i and n i for the ith separate model are obtained by using HS algorithm.The modeling design procedures are shown in Scheme 2.

Scheme 2. Harmony Search Based JPOC model design procedures.
Step 1. Choose single forecasting model and calculate separate forecasting results.Before the HS-based JPOC model is set up, the single forecasting model should be first selected according to the practical problem.For each model, the corresponding separate forecasting results can be calculated.
Step 2. Establish the joint parameters optimization combination (JPOC) model.Based on the single forecast, the JPOC combination model can be built up according to Equation (1).
Step 3. Determine the optimal values of the joint parameters ù i and n i by using the HS algorithm.
Step 4. Obtain the combination forecasting results from the HS-based JPOC model.

Data Sources
This section describes how to apply the HS algorithm to searching for the optimal values of exponential parameters and the combination forecasting weights and then establish the HS-based JPOC forecasting model.The yearly power generation data (Terawatt-hours, TWh for short) for China, Japan, Russian Federation and India from 2000 to 2010 obtained from the website of British Petroleum (BP) [30] were collected to validate the aforementioned method.The BP Statistical Review of World Energy which is one of the most widely respected and authoritative publications in the field of energy economics, provides high-quality, objective and globally consistent data on world energy markets.In 2010, the power generation for China, Japan and India accounted for 19.7%, 5.4% and 4.3% of the total power generation in the World, respectively.Combined, these top three countries constitute 29.4% of the global power generation and 76.36% of Asian power generation.The power generation in the Russian Federation accounts for 4.9% of the World total and nearly one fifth in the total of Europe and Eurasia in 2010.
Since China's reform and opening-up policy in 1980s, the average annual growth rate of GDP has been about 10%.Rapid and sustainable development of the economy has led to increased power generation, thus the power generation has grown rapidly from 300 TWh in 1980 to 4604 TWh in 2011.Now, the installed capacity ranks second in the World and the power generation ranks the first.Furthermore, the power generation in China will remain at high speed for decades since China is just in the process of industrialization and urbanization.
Japan is a country with rapid economic development.The Japanese economy has experienced a period of post-war economic recovery and rapid economic development in nineteenth century.And Japan has also experienced an atrophy period since the first 10 years of the 21st century.The power generation in Japan also shows fluctuating trends in typical years.How to accurately forecast power generation is difficult due to this fluctuating-type growth.
For a long time, India's power generation has found it difficult to meet the lighting needs of the residents and industrial electricity consumption due to the rapid economic development in the nation.Especially, blackouts have affected northern India, eastern and northeastern regions since 30 July, 2012.After years of rapid economic growth, electricity supply has become the bottleneck constraining growth.It is reported that during the 12th Five-Year Plan, India will make efforts to develop its power industry since 2012.The increase of India's power generation will accelerate in the future.To forecast future power generation in India has important theoretical and practical guiding significance.
Since the first eight years of the 21st century, the Russian Federation has experienced rapid economic growth.During the same period, the power generation also showed significant growth trends.Since September 2008, with the rapid spread of the international financial crisis and the global real economy downturn, Russia's economy fell into a severe recession.Therefore, the corresponding power generation decreased in 2009, so the power generation sequence of the Russian Federation shows a rising trend with typical fluctuations in certain years.
The yearly power generation curve (shown in Figure 1) exhibits different trends.The power generation curves of China and India show obvious rising trends, while the curves of Japan and Russian Federation show a basic rising trend with several waves.These four countries are selected as samples to test the applicability of the proposed HS-based JPOC forecasting model.Due to the different trends of the power generation curves, it is particularly meaningful to make accurate predictions.In next section, the performance data is presented to validate the aforementioned method.

Empirical Simulation
We conduct the experiments following the steps previously shown in Section 2.3.Firstly, we choose a separate forecasting model and calculate the single forecasting result.Linear regression model [31], time series model [32], Grey (1, 1) forecasting model (GM) [33] and Grey Verhulst model (GV) [34] are selected to generate the single forecasting result.Secondly, we establish the HS-based JPOC forecasting model according to Equation (1).Thirdly, determine the optimal value of the joint parameters using the HS algorithm.
A flowchart of the HS algorithm for parameter initialization is shown in Scheme 1.The details of the selection initial parameter model are as follows: HMS = 20, HMCR = 0.99, PAR = 0.5, BW = 1, lb = −100, ub = 100.All the programs were run on a 2.27 GHz Intel Core Duo CPU equipped with 1 GB of random access memory.In each case study, 30 independent runs were made for the HS optimization method in MATLAB 7.6.0(R2008a) under the 32-bit Windows 7 operating system.
The proposed HS-based JPOC model was validated with the power generation data from 2000 to 2010 for China, Japan, Russian Federation and India.Table 1 shows the optimal values of exponential parameters for the separate model and the combination forecasting weights for the four countries.The combination forecasting weights have both positive and negative values, as can be seen from Table 1.In combination forecasting, different single models play different roles in the combination model.There may be positive or negative correlations between the individual forecasting result series and the original data sequence, so the case that combination forecasting weights have positive and negative values is consistent with the actual situation.The combination forecasting weights adopted in this paper not only have positive and negative values, but also have no restriction that the sum of weights equals to 1.This forecasting weights processing method can achieve more accurate results.The forecasting values and the actual data for these countries are listed in Table 2. To test the forecasting performance, the HS-based JPOC model was compared with other four single models

Comparison with Four Other Single Models
This section focuses on the comparison between the HS-based JPOC model and the other four single models mentioned in this study.Table 3 and Figure 2 list the results of the HS-based JPOC model (HSC shown in figures), linear regression, time series, GM and GV forecasting models for China and the corresponding errors of these models.Due to the simple rising trend in China's power generation, the four separate models all capture the increasing trend better.The performance disparity for these five models can be identified from the errors in Table 3.For short range forecasting, the error range [−3%, +3%] is generally considered as a standard to measure forecasting result [35].Next, this range is adopted to compare the five methods as follows: the proposed HS-based JPOC model has only one forecasting result point that exceeds the range in a total of 11 points −4.0274% in 2007).The maximum and minimum errors are 2.0554% and −4.0274% in 2009 and 2007, respectively.In the regression model, there are four result points larger than 3%, two smaller than −3%, and two points near −3%, so in total six points are not satisfactory.The regression model reaches the maximum error of 5.7904% in 2003 and the minimum error of −15.3452% in 2000.In the time series model, there are two result points larger than 3%, one point smaller than −3%, and two points near −3%.The maximum error is 6.3466% in 2008 and the minimum error is −3.1404% in 2003.In GM mode, there are four result points larger than 3%, and two smaller than −3%.The maximum error is 6.6759% in 2002 and the minimum error is −6.3135% in 2007.In GV mode, there are three result points larger than 3%, one smaller than −3%.The maximum error is 6.2340% in 2002 and the minimum error is −3.8098% in 2007.Compared with the four single models, the numbers that exceed the error range for the HS-based JPOC model are the least, and the maximum and minimum errors are smaller than other single models.Table 4 lists the forecasting values and actual data of power generation for Japan and the corresponding errors.Figure 3 shows the curves of actual data and the forecasting results of the proposed model and the other four single models.The error analysis of Japan is as follows: In time series model, there are two points larger than 3%, four points smaller than −3%.The maximum error is 4.7080% in 2001 and the minimum error is −5.5917% in 2008.In GM mode, there is one result point larger than 3%, three smaller than −3%, and two points near +3%.The maximum error is 4.6903% in 2009 and the minimum error is −3.3175% in 2007.In GV mode, there are three points smaller than −3% and two points near +3%.The maximum error is 2.9643% in 2001 and the minimum error is −4.1962% in 2007.From errors analysis, we also conclude that the proposed model has better forecasting performance.For Japan's power generation sequence, there are two turning points (in 2001 and 2009).The forecasting errors of the proposed model for these two points are smaller than that of other single forecasting models which can be seen from Table 4.We can conclude that the HS based JPOC model can obtain better predictive performances in obvious turning points.
For the Russian Federation, no error result point of the proposed model exceeds the error range [−3%, +3%] (Table 5, Figure 4).There is only one result point larger than +3% or smaller than −3% for the linear regression, time series, GM and GV models, respectively.For India, we can also see that the errors of the result points are all within the [−3%, +3%] error range for the proposed model, time series model, GM model and GV model (Table 6, Figure 5).In the linear regression model, there is one result point larger than 3% and one point smaller than −3%.It seems that the proposed HS-based JPOC model does not display any obvious advantage concerning forecasting error range compared with other four single models, but from another point of view, we can analyze the maximal absolute percentage error (MaxAPE) indicator for these models.The MaxAPE indicator is defined as follows: max 100, 1, 2, , where t y is the power generation value in the tth year; t y  represents its forecasting result for the same period; and T is the number of data used for the MaxAPE calculation.For the Russian Federation, the MaxAPE values are 1.6544%, 3.5122%, 3.4231%, 3.6240% and 3.2144% for the proposed HS-based JPOC model, linear regression model, time series model, GM model and GV model respectively.For India, the MaxAPE values are 2.6794%, 5.3773%, 2.3000%, 2.0921% and 2.7272% for the corresponding models.Compared with other four single models, the MaxAPE of the HS-based JPOC model is smaller, which means the proposed model has less forecasting risk.For the Russian Federation's power generation sequence, there is also a turning point in 2009, and forecasting error of the proposed model is also smaller than the other single models.It is also tested by this case that the HS-based JPOC model can treat the sudden turning points better than other models.
Next, the mean absolute percentage error (MAPE) is adopted as an indicator of forecasting precision listed in Table 7.The calculation of the MAPE indicator was mentioned above in Equation (2).Among these five forecasting models, the HS-based JPOC model is the most accurate forecasting model because of its smallest MAPE value.Taking the MAPE of the HS-based JPOC model as a benchmark, the improvement rate with respect to the other four single models is also reported in Table 7.The improvement rates of regression, time series, GM and GV are 262.5863%,113.3657%, 167.7741% and 123.5114%, respectively, for China; 198.2371%, 291.8114%, 199.6679% and 191.7476% for Japan; 133.5178%, 136.3315%, 121.4022% and 130.9194% for the Russian Federation; 309.2960%, 138.8565%, 43.5823% and 144.7102% for India, respectively.Most MAPE improvements are over 100% for these four cases.The at least 43.5823% improvement reveals the superior forecasting performance of HS-based JPOC model.Therefore, it can be concluded that the proposed model is significantly more accurate than other four single forecasting models.The forecasting performance of the HS-based JPOC model (HSC shown in figures) is compared with four other combination models (EW, VACO, R, and DMFSE).In the DMSFE combination forecast model, the discounting factor β is chosen as 0.5.Table 8 lists the combination forecasting values and actual data of China's power generation and the corresponding errors between the actual value and the forecasting results.Figure 6 shows the curves of actual data and the forecasting results of the proposed model and the other four combination models.We can hardly observe the advantages of our proposed model from Figure 4 since these combination forecasting results are all very close to the actual values, so we also adopt error analysis for the proposed model and the other combination models.In the EW combination model, there are two result points larger than 3%, two smaller than −3%, and two points near 3%, so total of four points are not satisfactory.The EW model reaches the maximum error of 4.3410% in 2002 and the minimum error of −4.5508% in 2000.In the VACO combination model, there are two result points larger than 3%, two points smaller than −3%, and two points near 3%.The maximum error is 4.5242% in 2002 and the minimum error is −3.7388% in 2002.In the R combination model, there is one result point larger than 3%, and one smaller than −3%.The maximum error is 3.3169% in 2009 and the minimum error is −3.3755% in 2007.In the DMFSE model, there are two result points larger than 3%, two points smaller than −3% and two points near 3%.The maximum error is 4.3301% in 2002 and the minimum error is −4.4416% in 2000.In all, the numbers that exceed the error range for the HS-based JPOC model are the least, and the maximum and minimum errors are all smaller than those of the other combination models.The HS-based JPOC model showed better forecasting performance compared with the four other combination models for China.
Table 9 lists the forecasting values and actual data of power generation for Japan and the forecasting errors.Figure 7 shows the curves of actual data and the forecasting results for the five models.The error analysis of Japan is as follows: the proposed model has only one forecasting result point that exceeds the range (4.1535% in 2009).The minimum and maximum errors are 50.0121%and 4.1535% in 2005 and 2009, respectively.In the EW combination model, there are two result points larger than 3%, and four smaller than −3%.The maximum error is 3.2740% in 2001 and the minimum error 54.0454% in 2007.In the VACO combination model, there are two result points larger than 3%, two points smaller than 53%.The maximum error is 3.5009% in 2009 and the minimum error is 53.8751% in 2007.In the R combination model, there is only one result point larger than 3%.The maximum error is 3.6221% in 2009 and the minimum error is −1.6965% in 2010.In the DMFSE The forecasting results and errors for the Russian Federation and India are listed in Table 10 and Table 11.The curves of actual data and the forecasting results for the two countries are drawn in Figure 8 and Figure 9.No error point of the proposed model exceeds the range [−3%, +3%] for the Russian Federation.One result point is larger than +3% and one point is smaller than −3% for the EW model, the VACO model and the DMFSE model, respectively.Only one point is larger than +3% for the R model.For India, the errors of the four combination models are all within the error range.The HS-based JPOC model does not show any obvious advantage when dealing with the time series data trends of the Russian Federation and India.Next, we measure the forecasting risk by using the MaxAPE indicator.For the Russian Federation, the MaxAPE values for the five models are 1.6544%, 3.2021%, 3.2152%, 3.6079% and 3.2001%, respectively.The MaxAPE of the HS-based JPOC model is the smallest, which means that it will be less risky to choose the proposed model to forecast future trend.For India, the MaxAPE values of the five models all fluctuate around 2% (2.6749%, 2.3840%, 2.05491%, 1.7533% and 1.8513%).The MaxAPE of the HS-based JPOC model is not the best in this case, but through analyzing the absolute value of errors, we can find that only in 2000 and 2009, the errors showed worse results (1.5676% in 2000 and 1.6544% in 2009).Only two error points are slightly larger for India.Since the overall error indicator MAPE is adopted for the objective function, there may be certain individual points with slightly larger errors during the HS optimizing training process, but in other year points, the errors of the HS-based JPOC model are much smaller than those of the other combination models.The errors of the HS-based JPOC model expressed smaller fluctuations, which in not the case for the other combination models.Furthermore, the overall MAPE indicator is the smallest, which explains the comprehensive performance of the proposed model shown in Table 12.

Conclusions
It is well recognized that no single model consistently performs well in all situations.The combination model can always improve the accuracy of forecasting and is typically a reliable forecasting method for any practical forecasting issue.In this paper, the Harmony Search algorithm-based joint parameters optimization combination model is proposed for power generation forecasting.The single forecasting model adopts a power function form.The exponential parameters of the single power function model and the combination forecasting weights are then optimized simultaneously through using the HS algorithm to get the optimal parameter values.The combination forecasting results can be obtained finally.The yearly power generation data from 2000 to 2010 for typical countries with different trends are forecasted to test the effect and accuracy of the proposed method.Compared with four single models and four combination models for these four countries, the main conclusions drawn from the above study can be summarized as follows: first, the proposed combination model outperforms other single models and combination models for China, Japan, the Russian Federation and India.The numbers that exceed the error range [+3%, −3%] for the proposed model are the least, and the maximum and minimum errors are all smaller than other single models and combination models.Second, in terms of prediction accuracy, the proposed model is superior to other single models and combination models because it has the minimum MAPE value.Third, the proposed combination model could achieve better predictive performances at obvious turning points of power generation time series which can be reflected in several special points of the Japan and Russian Federation data.Even if there may be certain fluctuations in the future trends for power generation sequences, the proposed model could show promising results.In summary, all of those results showed that the proposed combination model is superior to the single models and other combination models for the test countries in terms of forecasting accuracy and model selection risk.

Figure 1 .
Figure 1.Yearly power generation in China, Japan, Russian Federation and Indian from 2000 to 2010 (TWh).
(linear regression model, time series model, GM model and GV model) and four combination models [Equivalent Weight (EW) model, Variance-Covariance (VACO) model, Granger and Ramanthan regression combination (R) model and Discounted MSFE model (DMSFE, â = 0.5)].The comparison results are shown in the next section.

Figure 2 .
Figure 2. Forecasting performance of HS based JPOC model and other four single models for China.

Figure 3 .
Figure 3. Forecasting performance of the HS-based JPOC model and four other single models for Japan.

Figure 4 .
Figure 4. Forecasting performance of HS based JPOC model and other four single models for Russian Federation.

Figure 5 .
Figure 5. Forecasting performance of the HS-based JPOC model and the other four single models for India.

Figure 6 .
Figure 6.Forecasting performance of the HS-based JPOC model and the other combination models for China.

Figure 7 .
Figure 7. Forecasting performance of HS based JPOC model and other combination models for Japan.

Figure 8 .
Figure 8. Forecasting performance of the HS-based JPOC model and the other combination models for the Russian Federation.

Figure 9 .
Figure 9. Forecasting performance of the HS-based JPOC model and the other combination models for India.

Table 1 .
The optimal value of ù i and n i for four countries.

Table 2 .
Forecasting results of the HS-based JPOC model for four countries (TWh).

Table 3 .
Forecasting results of HS based JPOC model and other four single models for China (TWh).

Table 5 .
Forecasting results of the HS-based JPOC model and other four single models for Russian Federation (TWh).

Table 7 .
The MAPE comparison of the HS based JPOC model and the other single models (%).

Table 8 .
Forecasting results of HS based JPOC model and other combination models for China (TWh).

Table 12 .
The MAPE comparison of the HS-based JPOC model and the other combination models (%).