Recurring Multi-layer Moving Window Approach to Forecast Day-ahead and Week-ahead Load Demand Considering Weather Conditions

Dao H. Vu; Kashem M. Muttaqi; Ashish P. Agalgaonkar; Arian Zahedmanesh; Abdesselam Bouzerdoum

网刊加载中。。。

使用Chrome浏览器效果最佳，继续浏览，你可能不会看到最佳的展示效果，

确定继续浏览么?

复制成功，请在其他浏览器进行阅读

Recurring Multi-layer Moving Window Approach to Forecast Day-ahead and Week-ahead Load Demand Considering Weather Conditions PDF

- ORCID：
Dao H. Vu (Student Member, IEEE)
✉
- ORCID：
Kashem M. Muttaqi (Senior Member, IEEE)
✉
- ORCID：
Ashish P. Agalgaonkar
✉
- ORCID：
Arian Zahedmanesh (Student Member, IEEE)
✉
- ORCID：
Abdesselam Bouzerdoum (Senior Member, IEEE)
✉

the School of Electrical Computer and Telecommunications Engineering, University of Wollongong, Wollongong, Australia； the Information and Computing Technology Division, College of Science and Engineering, Hamad Bin Khalifa University, Doha, Qatar； the School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, Wollongong, Australia

Updated：2022-11-19

DOI：10.35833/MPCE.2021.000210

OUTLINE

Abstract

The incorporation of weather variables is crucial in developing an effective demand forecasting model because electricity demand is strongly influenced by weather conditions. The dependence of demand on weather conditions may change with time during a day. Therefore, the time stamped weather information is essential. In this paper, a multi-layer moving window approach is proposed to incorporate the significant weather variables, which are selected using Pearson and Spearman correlation techniques. The multi-layer moving window approach allows the layers to adjust their size to accommodate the weather variables based on their significance, which creates more flexibility and adaptability thereby improving the overall performance of the proposed approach. Furthermore, a recursive model is developed to forecast the demand in multi-step ahead. An electricity demand data for the state of New South Wales, Australia are acquired from the Australian Energy Market Operator and the associated results are reported in the paper. The results show that the proposed approach with dynamic incorporation of weather variables is promising for day-ahead and week-ahead load demand forecasting.

Keywords

Autoregressive (AR) model; load forecasting; multi-layer moving window; Pearson correlation; Spearman correlation

I. Introduction

LOAD forecasting is crucial in an electricity network operation as it can provide critical information for not only maintaining the balance between the load and the generation, but also planning further expansion of the network. While over-forecast may lead to unnecessary generator commitments in the dispatch schedule, under-forecast could lead to purchases of expensive generators supplying peak demands [

1]. Consequently, more accurate forecast of electricity demand can help to harvest available generation economically and gain more benefits by operating the system efficiently. Furthermore, the reliability and quality of supply are enhanced significantly with the credibility of demand forecasting. For example, with an accurate demand forecasting, the generator scheduling can be precise thereby limiting unexpected and precarious load shedding events effectively.

A short-term load forecasting for individual residential households is proposed in [

2] using a long short-term memory recurrent neural network based framework. In [3], an autoregressive-based time varying (ARTV) model is developed for short-term forecasting of electricity demand. In this ARTV model, the associated coefficients are updated at a pre-set time interval following a pre-defined time-varying function. This function determines hourly and seasonal information which is included into the forecasting model, and these inclusions result in variability of the coefficients in different seasons, and finally lead to further improvement in the model performance without relying on any sub-models. While this model delivers an extraordinary performance for intra-hour resolution forecast, its performance in multi-step-ahead forecasting is particularly limited. When the forecasting horizon is extended to day- or week-ahead, the model performance may experience dramatic degradation because the correlation between demand and historical data depletes significantly, thus the use of sole historical data may not be sufficient. To minimize the degradation, exogenous weather variables should additionally be considered in the forecasting model [4].

The influences of climatic variables on electricity demand have been highlighted by numerous studies [

5]-[7]. This is due to the strong dependence of heating demand on the change of temperature [8], [9]. In [10], it is illustrated that the global warming can result in up to 6% additional demand consumption per year in Greece. In [11], it is shown that the change in peak electricity demand in Thailand may be increased by more than 15% at the end of the 21^st century. Although the effect of temperature on electricity demand is considered in [10] and [11], it is highlighted in [12] that the other climatic variables such as relative humidity and wind speed also have strong influence on electricity consumption. In [13], different dependent patterns of electricity demand on climatic variables in different seasons in the state of New South Wales (NSW), Australia are represented in four different regression models. It is shown that the per capita demand for NSW increases 6.14% and 11.3% in summer and spring, respectively, but decreases by 4.11% and 0.45% in winter and autumn, respectively, by the end of the 21^st century. In [14], it is reported that the sensitivity of climate change to electricity demand is different for four big cities of Australia, where 7 ℃ increase in temperature may cause 1.5% growth in yearly average demand of Sydney and Melbourne, and 10%-28% growth in yearly average demand of Brisbane and Adelaide. In [15], it is revealed that 1 ℃ increase in temperature will result in 1.4% drop in yearly average electricity demand. However, 1 ℃ decrease in temperature will result in 1.6% increase in yearly average electricity demand in New Zealand. In [16], it is indicated that the residential cooling demand is strongly impacted by the change in climatic variables, especially the temperature in the USA. Also, it is found that 20% increase in cooling degree days (CDDs) will lead to 1%-9% increase in total residential electricity demand and 20%-60% increase in residential AC electricity energy consumption. In [17], it is estimated that a 1.5 ℃ increase in temperature would increase total net expenditure by 4% for the state of California by the year 2100 due to increased cooling and decreased heating demands.

Due to the strong influence of weather variables on electricity demand [

18], it is necessary to incorporate weather variables in the demand forecasting model thereby improving the forecasting performance. In [19], temperature is included into a demand forecasting model using an epi-splines function. This function represents the relationship between demand and temperature by classifying temperature in each season into three different levels, i.e., low, middle, and high, and thus helps capture the non-weather dependent load pattern in the forecasting model. In [20], weather variables such as temperature and humidity are incorporated in demand forecasting model as exogenous variables using two models which are linear autoregressive model with exogenous variables and non-linear autoregressive model with exogenous variables. In [21], the multiple linear regression (MLR) model is used to account for the influence of temperature on electricity demand. This model represents the heat build-up effect which is due to the absorption and storing temperatures in building and surrounding environments, thus improving the demand forecasting performance. In [22], temperature is included into the demand forecasting model using kernel-based support vector quantile regression and Copula theory, which can improve the ability to estimate the probability density of the forecasting load in short-term period significantly. In [23], it is reported that the weather variables such as temperature can be employed to determine the load shape of demand which then can be used to forecast the day-ahead load effectively. A deep learning based model which refines features by stacked denoising auto-encoders (SDAs) from historical electricity demand data and related temperature parameters is proposed in [24]. It trains a support vector regression (SVR) based model to forecast the day-ahead electricity demand. An interval probability distribution learning (IPDL) model is proposed in [25] for wind speed forecasting based on restricted Boltzmann machines and rough set theory to capture unsupervised temporal features from the wind speed data. A deep belief network embedded with parametric Copula models is proposed in [26] to forecast day-ahead and week-ahead load demands with hourly resolution.

In the above-mentioned models, the incorporation of weather variables in the forecasting model is static; however, the impacts of temperature on demand may be dynamic. Consequently, if a dynamic inclusion of the weather variable is considered, it will result in more effective demand forecasting model. For example, a moving window model based on the autoregressive model with exogenous variables is used in [

28] to accommodate weather variables in demand forecasting. It is shown in [28] that the obtained model exhibits high accuracy of load forecasting in short-term period. This performance is predominant due to the hourly variation of coefficients of different sub-models. In other words, the dynamic variation of coefficients of different sub-models helps improve the performance of the forecasting model considerably. Also, the coordination of time stamp in conjunction with weather variables would be effective in deriving accurate forecast of demand [29].

From the research reported in the literature, it can be seen that some demand forecasting models do not include weather variables although they have strong influence on the electricity demand. This limits the performance of the proposed forecasting models. Realizing this limitation, some studies include weather variable variables in their models, but the dynamic characteristic of the weather variation is not fully addressed. There are many other considerations such as forecasting horizons, the significance of the exogenous variables, and the development of recursive algorithm, which have been thoroughly investigated through the proposed methodology. The main motivation behind the proposed work is to develop a dynamic model to forecast electricity demand in short-term period with high accuracy.

In this paper, a recurring multi-layer moving window approach is proposed, which includes three stages to guarantee the robust implementation of the load demand forecasting. At the first stage, two correlation techniques namely Pearson and Spearman correlation techniques are employed to analyze the impacts of weather variables on electricity demand. At the second stage, a multi-layer moving window approach is used to incorporate the predominant weather variables on the electricity demand forecasting model. The size of the layers is adjustable to accommodate the weather variables based on their significance. This creates more flexibility and adaptability to forecast the demand in different ranges. Meanwhile, it reduces the calculations since the layer size can be assigned based on the most recent and important roleplaying factors that affect the demand. At the third stage, a recursive structure is employed to forecast the day- and week-ahead load demand. This structure allows the forecasted values to be reused as inputs thus updating the model consecutively and improving the overall forecasting results.

The main contributions of the paper can be summarized as follows.

1) At the first stage, the designed approach pervasively analyses the impacts of available weather variables on electricity demand using Pearson and Spearman correlation techniques. Meanwhile, the significance of weather effects is estimated using p-values derived from the t-test.

2) A multi-layer moving window approach is proposed to incorporate the assigned weather variables at the first stage into the demand forecasting model. This approach allows adjustable layer size to accommodate the weather variables based on their significance, which offers more flexibility and adaptability thereby improving the model performance significantly.

3) A recursive model is developed to forecast the demand in multi-step ahead. This model allows the forecasted values to be reused before finalising the forecasting outcome. This provides more useful information and consequently, reduces the error of multi-step-ahead forecasting.

4) The electricity demand data for the state of NSW, Australia are acquired from the Australian Energy Market Operator (AEMO) for a case study analysis. The results show that the proposed approach is robust and effective. The day- and week-ahead demand forecasting results are promising.

The rest of the paper is organized as follows. Section II presents the impacts of weather variables on electricity demand. Section III introduces a forecasting approach which employs a single- and multi-layer moving window approach to include the weather variables and uses a recursive model to forecast the demand in multi-step ahead. Section IV highlights some experimental results and model validation, and Section V provides the concluding remarks.

II. Impacts of Weather Variables on Electricity Demand

The weather variables may have significant influence on the electricity demand. Among all the weather variables, temperature is reported to be the most important variable that can have significant impact on the electricity demand [

27]. The relationship between the load and the temperature and the effectiveness of the use of temperature for the electricity forecasting have been confirmed in many studies before [27]. In this regard, among all other weather variables, the temperature is the most pronounced weather variable to forecast the load demand [27], [28]. Additionally, the other weather variables such as irradiation may have linear relationship with the electricity demand. The data associated with number of weather variables for the monthly resolution are included and analyzed in our previous work [28], [30]. In the proposed study, the data with half-hourly resolution are employed. For this resolution, three main datasets which are temperature, humidity, and wind speed are available for further analysis. As a result, only these variables are analyzed in the proposed study. In case the other variables such as irradiation are available, their impacts on load demand can be analyzed in the similar manner.

A. Weather Variables and Electricity Demand

1)　Electricity Demand

Electricity demand data used in this study are available from the AEMO [

31] with a resolution of 30 min. The variation of the obtained data is shown in Fig. 1.

Fig. 1 Historical data of electricity demand for NSW, Australia.

It can be observed from Fig. 1 that the electricity demand encounters seasonal variation in both short- and long-term periods. Also, it is noted that the demand has decreasing trend from 2010 to 2015. This trend may be partially due to the growth of solar photovoltaic (PV) installation at residential household level, and this causes the deduction in the net energy consumption.

2)　Weather Variables

Weather variables, which represent atmospheric conditions, typically include temperature, humidity, and wind speed. These variables may experience considerable variation following calendar seasons in a year. In order to illustrate the variation of weather variables, a dataset has been acquired from the Sydney airport weather station at NSW, Australia [

32]. This dataset includes temperature, humidity, and wind speed with resolution of 30 min. The time series representation of each of these variables is given in Fig. 2(a)-(c), respectively.

Fig. 2 Historical data of weather variables at Sydney airport weather station at NSW, Australia. (a) Temperature. (b) Humidity. (c) Wind speed.

Figure 2(a) shows that the temperature has strong variation along the years with high values at the beginning and the end of each year, i.e., the months of January and December, respectively, and low values at the middle of the year, i.e., the months of June and July. In contrast to this strongly seasonal variation of temperature, the variations of humidity and wind speed are much more random with very weak seasonality as shown in Fig. 2(b) and (c), respectively. The characteristics of these typical weather variables are given in Table I.

Table I Characteristics of Temperature, Humidity, and Wind Speed

Characteristic	Temperature (℃)	Humidity (%)	Wind speed (km/h)
Minimum value	3.2	3.0	0
Maximum value	45.5	100.0	80.0
Mean value	18.5	66.7	20.3
Median value	18.7	68.0	18.0

It can be observed from Table I that the variations of all three variables are significant. The wind speed can vary from 0 (stalling state) to 80 km/h, and the relative humidity spreads out from 3.0% to 100%. Interestingly, the temperature is always positive with the minimum and maximum values being at 3.2 ℃ and 45.5 ℃, respectively. It is noted that the median value of humidity is higher than its mean value; however, the median value of wind speed is lower than its mean value. The comparison between mean and median values illustrates the skewness of the dataset, and thus majority of the humidity data are higher than its mean value of 66.7% while the majority of wind speed data are lower than its mean value of 20.3 km/h. In contrast, the mean and median values of temperature are very close to each other so that the temperature dataset may not experience extreme skewness.

The variabilities associated with temperature, humidity, and wind speed are shown in Fig. 3 considering smaller time resolution for a typical week in 2015. It is noted that the humidity may have negative correlation with temperature in the selected timeframe because temperature is at a low level when humidity is high, and vice versa. On the other hand, the variation of wind speed is random, and it hardly follows any specific pattern in the shorter timeframe.

Fig. 3 Hourly variation of temperature, humidity, and wind speed for a typical week in July 4-11, 2015.

B. Correlation Analyses

Weather variables may have strong influence on human feeling and living style. In order for humans to maintain the living comfortability, the adverse weather impacts can be reduced using modern electric equipment. As a result, small variations in weather variables can have enormous influence on an electricity consumption [

6], [7]. To reveal the impacts of weather variables on demand, correlation analysis can be used. The two most widely used correlation techniques are Pearson and Spearman correlation. While the former reveals the linear relationship, the latter discloses the monotonic relationship between the associated variables.

Considering the threshold value, the balance point that is the threshold value for the weather variables, namely, the temperature, humidity and wind speed, is specified based on the lowest electricity demand for the studied data in NSW, Australia. Particularly, the balance point or the threshold value for the temperature is normally about 18.30 ℃ and 21.00 ℃ for moderate and warm environments, respectively [

28]. Meanwhile, a “V-shaped” trend curve can be considered to demonstrate the relationship between the temperature and the load, in that case, e.g., to supply the heating, ventilation, and air conditioning (HVAC) systems, the demand increases when the temperature increases or decreases from the balance point.

1)　Pearson and Spearman Correlation Techniques

The Pearson correlation technique is commonly used to estimate the linear interdependency among different variables [

33]. Using this technique, a correlation coefficient, which is between -1 and +1, is generated to estimate the degree of correlation between different variables [34]. While +1 indicates a perfect positive correlation; -1 signifies a perfect inverse correlation; and 0 means no correlation. The Spearman correlation is used to evaluate the monotonic relationship between two variables. In a monotonic relationship, the variables tend to change together, but not necessarily at a constant rate.

2)　Significance of Correlation Coefficients

While the calculated correlation coefficients measure the strength of the relationship between two variables, the significance of the relationship or the correlation coefficients can be tested using the t-test proposed in [

35]. In this test, the data are assumed to follow the normal distribution and a test score is calculated using the number of observation and the correlation coefficient [35] as:

t = \frac{r \sqrt[]{n - 2}}{\sqrt[]{1 - r^{2}}}

(1)

where r is the correlation coefficient; and n is the number of observations.

Furthermore, a p-value (denoted as p_v), which represents a probability level indicating how unlikely a given correlation coefficient will occur given no relationship in the population, is estimated based on the obtained test score as:

p_{v} = 2 \cdot p r o (T > t)

(2)

where pro( $\cdot$ ) is the probability of an incident to occur; and T conforms to a normal distribution.

It is noted that the p_v also represents the probability of incorrectly rejecting a true null hypothesis. It is interpreted as an indicator of statistical significance of the correlation between two variables. For example, if a threshold value of 0.05 is considered, it is accepted that a true null hypothesis (there is no correlation between the two variables) may be incorrectly rejected with a probability of 5%. This threshold value is widely accepted in the literature as indicated in [

36]. Consequently, this value is selected to be the threshold in our study.

3)　Associated Results

The Pearson and Spearman correlation coefficients (denoted as $r_{x y}$ and $r s_{x y}$ ) and their significance values are calculated and the results are presented in Tables II and III, respectively. In order to evaluate the strength of correlation, the mean absolute values of weekly correlation coefficients in the entire dataset are considered.

Table II Pearson Correlation Coefficients and Their Significance Values Between Electricity Demand and Weather Variables

Variable	$r_{x y}$	p_v
Temperature	0.575	0.031
Humidity	0.158	0.108
Wind speed	0.118	0.159

Table III Spearman Correlation Coefficients and Their Significance Values Between Electricity Demand and Weather Variables

Variable	$r s_{x y}$	p_v
Temperature	0.570	0.006
Humidity	0.141	0.126
Wind speed	0.112	0.157

It can be observed from Tables II and III that the temperature has strong correlation with the electricity demand as the correlation coefficients are greater than 0.5. The two correlation coefficients are not much different from each other, inferring that the monotonic relationship between the temperature and electricity demand is dominated along with the linear relationship. In addition, the p_v values of the temperature are smaller than 0.05, which demonstrates that the linear correlation is significant at the level of 5%.

On the other hand, it can be observed from Table II that the Pearson correlation coefficients for the humidity and wind speed with electricity demand are much smaller than 0.25, illustrating the weak relationship between these two variables and electricity demand. Also, it can be observed from Table II that the p_v values of the humidity and wind speed are much greater than 0.05. Accordingly, these correlation coefficients are not significant at the level of 5%. Similarly, it can be observed from Table III that the Spearman correlation coefficients for the humidity and wind speed are weak and not significant at the level of 5%.

In the above analyses, only the temperature shows strong and significant correlation to the electricity demand. Consequently, only the temperature should be considered to be included in the forecasting model for the data acquired from NSW, Australia.

III. Demand Forecasting Incorporating Weather Variables

In this section, a new forecasting approach is proposed to build a demand forecasting model using a multi-layer moving window approach and a recursive model. While the multi-layer moving window approach is employed to incorporate weather variables, the recursive model is introduced for day- and week-ahead forecasting of electricity demand.

A. Forecasting Strategy

The conceptual diagram of the proposed forecasting approach is given in Fig. 4. Firstly, the historical data of demand and weather variables are employed to conduct the correlation analyses and the correlation results are used to select the significant variables to the demand forecasting model. Secondly, a multi-layer moving window approach is developed to include the selected weather variables into the demand forecasting model, which is based on a single-layer moving window approach. Thirdly, a recursive model is established to extend the forecasting ability to day- and week-ahead timeframes with a resolution of 30 min.

Fig. 4 Conceptual diagram of proposed forecasting approach.

B. Single-layer Moving Window Approach

In the single-layer moving window approach, a dataset is rearranged based on the observed patterns and the forecast can be represented in the form of links between sub-windows. A typical single-layer moving window with a size of $3 \times 3$ sub-windows is given in Fig. 5. Each small window in Fig. 5 represents a state for a typical single-layer moving window with a size of $3 \times 3$ sub-windows, where i represents the hour and j represents the day of the current state, respectively; and the grey color represents the previous state. Meanwhile, it is required to forecast the value at state $(i + 1, j)$ . To forecast this value, all the other values from (i,j) backward can be used.

Fig. 5 Diagrammatic representation of single-layer moving window approach.

It is noted from [

3] that the autoregressive model can be used to represent the link between forecasting points and historical data values. The autoregressive model can be expressed as:

y (k + 1) = a_{0} + \sum_{i = 1}^{m} a_{i} y (k - l_{i}) + e (k + 1)

(3)

where $y (k + 1)$ is the forecasting demand; $a_{i} (i = 0,1, \dots, m)$ is the coefficient; $y (k - l_{i})$ is the lag (l_i) of the current demand value $y (k)$ ; and $e (k + 1)$ is the error.

If (3) is used to represent the window in Fig. 5, $y (k + 1)$ signifies the sub-window $(i + 1, j)$ , and the historical sub-windows from $(i, j)$ to $(i + 1, j - 2)$ are denoted by $y (k - l_{i})$ , $i = 1,2, \dots, m$ .

C. Multi-layer Moving Window Approach

In the multi-layer moving window approach, additional layers will be included to signify the dependence of demand on other variables thereby improving the performance of the forecasting model. For example, if a weather variable $x$ contributes to the improvement of the model in (3), this variable can be used to model the residual of (3) as:

e (k + 1) = \sum_{i = 1}^{p} b_{i} x (k - h_{i}) + u (k + 1)

(4)

where b_i $(i = 1,2, \dots, p)$ is the coefficient; $x (k - h_{i})$ is the historical value of variable x; h_i is the respective hour; and $u (k + 1)$ is the residual.

Substituting (4) into (3) and extending the expression for different weather variables, the autoregressive-based forecasting model can be rewritten as:

y (k + 1) = a_{0} + W_{1} (k + 1) + W_{2} (k + 1) + . . . + W_{n} (k + 1) + ε (k + 1)

(5)

where $ε (k + 1)$ is the error term; and $W_{i} (k + 1)$ includes different layers of a moving window, which can be calculated as:

\{\begin{array}{l} W_{1} (k + 1) = \sum_{i = 1}^{m} a_{i} y (k - l_{i}) \\ W_{2} (k + 1) = \sum_{i = 1}^{p} b_{i} x (k - h_{i}) \\ ⋮ \\ W_{n} (k + 1) = \sum_{i = 1}^{q} c_{i} z (k - h_{i}) \end{array}

(6)

where c_i ( $i = 1, 2, \dots, q$ ) is the coefficient; and $z (k - h_{i})$ is the historical value of variable $z$ .

It is noted that (5) represents the summation of different window layers W_i, which is a numerical representation of multi-layer moving window. Graphically, the multi-layer moving window can be represented as in Fig. 6. Figure 6 shows that there can be n layers in the moving window and each layer represents one input variable. It is noted that the window size in each layer can be adapted to appropriately depict the significance of the input variables. Furthermore, the coefficients are adaptively altered to update the impacts of weather variables.

Fig. 6 Graphical representation of multi-layer moving window approach.

D. Recursive Forecasting Model

The recursive model is employed to forecast electricity demand in multi-step ahead. This model allows the newly forecasted value to be used as one of the inputs to forecast the demand in the consecutive forecasting steps.

It is noted that (5) is used to forecast one-step ahead only. To forecast s-step ahead, (5) can be extended as:

y (k + s) = a_{0} + W_{1} (k + s) + W_{2} (k + s) + . . . + W_{n} (k + s) + ε (k + s)

(7)

where y(k+s) is the s-step-ahead forecasting value.

In (7), there are two main types of windows, namely, internal and external windows. The external window includes W_i(k+s) layers, where $i \geq 2$ , which are independent from the historical demand thus can be updated accordingly. On the other hand, the internal window, i.e., the first window layer W₁(k+s), is associated with the historical data, so that it can only be updated using a recursive structure. The recursive structure allows the forecasting process to consider the forecasting value at a previous step as one of the inputs to the model to forecast the subsequent value. For illustration, the representation of two-step-ahead forecasting is given as:

W_{1} (k + 2) = \sum_{i = 2}^{m + 1} a_{i} y (k - l_{i}) = a_{1} {\hat{y}}_{m + 1} + \sum_{i = 2}^{m} a_{i} y (k - l_{i})

(8)

where ${\hat{y}}_{m + 1}$ is the estimated/forecasted demand.

Similarly, the s-step-ahead forecasting can be achieved by expanding (8) into (9).

W_{1} (k + s) = \sum_{j = 1}^{s - 1} a_{j} {\hat{y}}_{m + j} + \sum_{i = s}^{m} a_{i} y (k - l_{i})

(9)

In (9), while the second part $\sum_{i = s}^{m} a_{i} y (k - l_{i})$ represents the historical dependence, the first part $\sum_{j = 1}^{s - 1} a_{j} {\hat{y}}_{m + j}$ represents the previous forecasting dependence. If s is greater than m, the second part of (9) is not required to be considered. Therefore, from the m-step-ahead forecasting, the forecasting outcome is only dependent on the previously obtained forecasting values.

The problem of error accumulation is expected to be not significant for the proposed approach. This is because normally the forecasting value at a previous step is the actual value for the electricity load, and clearly this won’t introduce any model’s error to forecast the subsequent value.

IV. Results and Discussion

A case study has been conducted with the aid of electricity demand data, acquired from AEMO, for the state of NSW, Australia to demonstrate the effectiveness and robustness of the forecasting model developed in this paper. Also, the forecasting result from this model is compared with three different benchmark models for the validation purpose.

It is noted that the size of the window indicated in Figs. 5 and 6 depends on the number of past datasets to be used. The inclusion of more past datasets in the window increases the computational burden due to involved numerical calculations for every step of the iteration, which may not be effective and efficient for real-time applications due to data latency. In this study, which is aimed to be used in real time, the optimal window size considering the above constraints is $3 \times 3$ .

A. Testing Procedure

The data used in this study are available from AEMO [

31] with resolution of 30 min. The whole dataset from 2010 to 2015 is divided into two main datasets which are training datasets (from 2010 to 2014) and testing/forecasting dataset (2015) for training and testing purposes, respectively. The training dataset is used to train the proposed model, and then the devised model is tested on the forecasting dataset. The training and forecasting processes are repeated for all considered models for a comparative study with the consideration of the same input variables including the consecutive time lags of demand and temperature (representing the window size of

3 \times 3

), hour, weekday, and holiday indicator (1 if it is holiday, and zero otherwise).

To evaluate the performance of the proposed model, various error measures can be employed. In the literature, three error measures which are root mean square error (RMSE), mean absolute percentage error (MAPE), and coefficient of determination R², are widely used in demand forecasting evaluation. These measures are employed in this study, and the calculations of them are given as:

M A P E = \frac{100}{N} \sum_{i = 1}^{N} \frac{|y_{i} - {\hat{y}}_{i}|}{y_{i}}

(10)

R M S E = \sqrt[]{\frac{1}{N} \sum_{i = 1}^{N} (y_{i} - {\hat{y}}_{i})^{2}}

(11)

R^{2} = 1 - \frac{\sum_{i = 1}^{N} (y_{i} - {\hat{y}}_{i})^{2}}{\sum_{i = 1}^{N} (y_{i} {- \bar{y})}^{2}}

(12)

where y_i is the actual demand; ${\hat{y}}_{i}$ is the forecasted demand; $\bar{y}$ is the mean value of actual data; and N is the number of the forecasted data points.

Furthermore, the testing results are employed to investigate the significant contribution of weather variables to the accuracy of the demand forecasting model. In order to examine this contribution, the proposed multi-layer moving window model is compared with the single-layer moving window model, in which only historical demand data are considered, as discussed in Section III-B. The two models are investigated in two main forecasting horizons which are day-ahead and week-ahead.

B. Day-ahead Forecasting

In this sub-section, the forecasting is conducted for one-day ahead forecasting. The model is trained with data from 2010 to 2014, and the trained model is tested on the dataset (2015). In order to visualize the performance of the model based on the proposed approach, the forecasting results in one typical week, i.e., June 6-13, 2015, are extracted and presented in in Fig. 7.

Fig. 7 Day-ahead demand forecasting using model based on proposed approach and single-layer moving window model in June 6-13, 2015.

Figure 7 shows that the forecasting result from the model based on the proposed approach has closer correspondence with the actual demand than that from the single-layer moving window model. Looking more closely at the forecasting result for June 7, 2015, it can be observed that the model based on the proposed approach outperforms the single-layer moving window model. In other words, the forecasting results are improved considerably with the incorporation of weather variables. To reveal the impacts of weather variables, e.g., temperature, in the tested timeframe, the temperature variations on June 6 and 7, 2015 are compared with the temperature profile from the previous week and the results are given in Fig. 8.

Fig. 8 Variation of temperature on June 6 and 7, 2015.

It can be observed from Fig. 8 that the temperature on June 6, 2015 is nearly the same as on the same day in the previous week. Contrarily, the temperature on June 7, 2015 encounters significant variations compared with that on the same day in the previous week. The associated temperature difference is up to 10 ℃ during the daytime. This increment in temperature leads to considerable reduction in heating requirement and thus moderates the electricity demand. Consequently, the demand is much lower on June 7, 2015 compared with that on the same day in the previous week. The impact of temperature has been clearly reflected in the forecasting results of the model based on the proposed approach. Hence, the accuracy of the model based on the proposed approach is significantly higher than that of the single-layer moving window model as presented in Fig. 7.

To evaluate the performances of the model based on the proposed approach and the single-layer moving window model for day-ahead forecasting, MAPE values are calculated and presented in Fig. 9.

Fig. 9 MAPE values for day-ahead forecasting using model based on proposed approach and single-layer moving window model.

Figure 9 shows that the MAPE values of both models experience a strong increase when the forecasting horizon increases from half-hour-ahead to day-ahead. This indicates that both models experience significant deterioration due to the inherent uncertainty of load demand with multi-step- ahead forecasting. In addition, it can be inferred from the MAPE values in Fig. 9 that the model based the proposed approach outperforms the single-layer moving window model.

Furthermore, the difference between the MAPE values for the two models is more significant at larger forecasting horizon. While this difference is only 0.01% for half-hour-ahead forecasting, it is about 0.8% for day-ahead forecasting. This signifies that the weather variables play a key role in demand forecasting at larger forecasting horizon.

C. Week-ahead Forecasting

In this sub-section, the model based on the proposed approach and the single-layer moving window model are employed to forecast electricity demand one-week ahead. The forecasting performance of these two models are compared and shown in Fig. 10.

Fig. 10 MAPE values for week-ahead forecasting using model based on proposed approach and single-layer moving window model.

Figure 10 shows that when the forecasting horizon is extended from day-ahead to week-ahead, MAPE values of both models experience slight increases. While the increase is about 1.6% (from nearly 3.5% to around 5.1%) for the single-layer moving window model, it is about 0.9% (from 2.8% to 3.7%) for the model based on the proposed approach. These increases are significantly smaller when forecasting horizon is extended from half-hour-ahead to day-ahead.

It is noted from Fig. 10 that the MAPE values may follow a logarithmic function with an upper limit when the forecasting horizon is extended. This demonstrates the robustness of the model based on the proposed approach.

D. Error Dispersion Analysis

The model based on the proposed approach is employed to forecast the electricity demand for NSW, Australia for the entire year of 2015. The dispersion of obtained forecasting errors is investigated in different hours of a day and different days of a week. To reveal these dispersions of the error, the box plot is used and the results are given in Figs. 11 and 12.

Fig. 11 Forecasting error for different hours of a day.

Fig. 12 Forecasting error for different days of a week.

It can be observed from Fig. 11 that the error is smaller in the early morning and at late night of the day. More specifically, while the error is quite small in the period between 23:00 to about 09:00, it may be considerable in the period from 10:00 to 22:00. In addition, the error can stretch to peak values in the two sub-periods, which are 06:00 to 10:00 and 15:00 to 18:00.

Figure 12 shows that some outliers are more dispersed for Thursday than those for Wednesday while the whiskers of the box are closer to zero for Thursday than for Wednesday. This may result in smaller MAPE but higher RMSE as the weights for the outliers are squared in RMSE calculation. While MAPE focuses more on average error, RMSE pays more attention on the outliers. Therefore, both measures have been considered in evaluating the forecasting results. The MAPE and RMSE for different days of a week are given in Table IV.

Table IV Proposed Model Performance in Different Days of a Week

Time	MAPE (%)	RMSE (MW)	R²
Monday	3.165	384	0.910
Tuesday	3.192	379	0.899
Wednesday	2.830	329	0.928
Thursday	2.765	350	0.918
Friday	3.046	384	0.899
Saturday	2.963	361	0.853
Sunday	2.739	301	0.919

Table IV shows that while the forecasting errors are small for Wednesday, Thursday, Saturday and Sunday with MAPE value being lower than 3.0%, it is more significant for the other days of the week with MAPE value being greater than 3.0%. The determination coefficient R²in Table IV shows that the forecasting values fit very well with the actual values. This demonstrates the effectiveness and robustness of the model based on the proposed approach.

E. Comparative Studies

In order to validate the proposed model, other load forecasting models including naïve model, autoregressive model, neural network model, and single-layer moving window model are employed as benchmark models for the comparison purpose. The comparison is conducted for day-ahead and week-ahead forecasting. The comparative results of MAPE values are given in Table V.

Table V Comparative Results of MAPE Values for Day-ahead and Week-ahead Forecasting

Model	MAPE (%)
Model	Day-ahead	Week-ahead
Naïve model	4.964	4.967
Autoregressive model	3.561	5.268
Neural network model	3.467	4.794
Single-layer moving window model	3.464	5.245
Model based on proposed approach	2.830	3.833

It can be observed from Table V that the model based on the proposed approach outperforms the other benchmark models in various forecasting horizons including day-ahead and week-ahead forecasting. The day-ahead forecasting result of Table V illustrates that the model based on the proposed approach significantly outperforms the other models for day-ahead forecasting. Specifically, the MAPE value of the model based on the proposed approach is estimated to be 2.830%, which is just more than half of that from the naïve model and is about 20% lower than that of the neural network model. Furthermore, it can be observed from the week-ahead forecasting result that the model based on the proposed approach shows a promising result with the MAPE value being 3.833%, which is the lowest in the five considered models. The results in Table V clearly demonstrate the advancement of the model based on the proposed approach against its counterparts including neural network and autoregressive models for both day-ahead and week-ahead forecasting.

Although the machine learning (ML) approaches, e.g., artificial neural network (ANN) and SVR, can also show effective performance in some aspects, e.g., to exploit the non-linear relationship of load demand and other variables, they have the following drawbacks: ML models are normally non-flexible models that are based on highly complex commercial systems, as such, they are regarded as black-box approaches where the mathematics of the system is not available in detail. This is particularly problematic when the system is required to be adjusted by the network users for different conditions. The other disadvantage is that the implementation of ML models requires advanced calculation packages with complex structure [

27].

Unlike the ML models, the proposed approach offers flexibility and adaptability to the user control, e.g., the window size of the variables’ layer can be changed based on the significance of variables under different regional conditions. Moreover, the proposed approach can be implemented with uncomplicated structure and minimized mathematical calculations, and guarantees the robust performance to capture the variation of weather variables for the day-ahead and week-ahead forecasting.

V. Conclusion

In this paper, the patterns of typical weather variables including temperature, humidity, and wind speed are revealed using time series plots. Furthermore, the impacts of these variables on electricity demand are analyzed using correlation techniques. The results from these analyses illustrate that the demand is significantly influenced with the variation of weather variables, especially temperature.

Since weather variables may dynamically be associated with an electricity demand, a multi-layer moving window based model is developed to incorporate the influence of weather variables in demand forecasting. Each layer of the window can represent a weather variable effectively because the individual layer can adjust the size based on significance of each weather variable. The model is then extended to forecast the day-ahead and week-ahead demands using a recursive technique. This technique allows the forecasting value to be reused as an input in the model, and hence the overall performance of the forecasting model has been improved further. A typical dataset for the state of NSW, Australia is used to illustrate the effectiveness of the proposed approach, which can pave the way for state-wide demand management strategy in future. The obtained results demonstrate the significance of weather variables in electricity demand forecasting. Dynamic incorporation of weather variables in the model based on the proposed approach helps improve the performance of the multi-step-ahead forecasting significantly.

References

D. K. Ranaweera, G. G. Karady, and R. G. Farmer, “Economic impact analysis of load forecasting,” IEEE Transactions on Power Systems, vol. 12, no. 3, pp. 1388-1392, Aug. 1997. [Baidu Scholar]

W. Kong, Z. Dong, Y. Jia et al., “Short-term residential load forecasting based on LSTM recurrent neural network,” IEEE Transactions on Smart Grid, vol. 10, no. 1, pp. 841-851, Jan. 2019. [Baidu Scholar]

D. H. Vu, K. M. Muttaqi, A. P. Agalgaonkar et al., “Short-term electricity demand forecasting using autoregressive based time varying model incorporating representative data adjustment,” Applied Energy, vol. 205, pp. 790-801, Nov. 2017. [Baidu Scholar]

H. Takeda, Y. Tamura, and S. Sato, “Using the ensemble Kalman filter for electricity load forecasting and analysis,” Energy, vol. 104, pp. 184-198, Jun. 2016. [Baidu Scholar]

P. Jiang, F. Liu, and Y. Song, “A hybrid forecasting model based on date-framework strategy and improved feature selection technology for short-term load forecasting,” Energy, vol. 119, pp. 694-709, Jan. 2017. [Baidu Scholar]

T. K. Mideksa and S. Kallbekken, “The impact of climate change on the electricity market: a review,” Energy Policy, vol. 38, no. 7, pp. 3579-3585, Jul. 2010. [Baidu Scholar]

R. Schaeffer, A. S. Szklo, A. F. P. de Lucena et al., “Energy sector vulnerability to climate change: a review,” Energy, vol. 38, no. 1, pp. 1-12, Feb. 2012. [Baidu Scholar]

M. Noussan, M. Jarre, and A. Poggio, “Real operation data analysis on district heating load patterns,” Energy, vol. 129, pp. 70-78, Jun. 2017. [Baidu Scholar]

I. M. Trotter, T. F. Bolkesjø, J. G. Féres et al., “Climate change and electricity demand in Brazil: a stochastic approach,” Energy, vol. 102, pp. 596-604, May 2016. [Baidu Scholar]

S. Mirasgedis, Y. Sarafidis, E. Georgopoulou et al., “Modeling framework for estimating impacts of climate change on electricity demand at regional level: case of Greece,” Energy Conversion and Management, vol. 48, no. 5, pp. 1737-1750, May 2007. [Baidu Scholar]

S. J. Parkpoom and G. P. Harrison, “Analyzing the impact of climate change on future electricity demand in Thailand,” IEEE Transactions on Power Systems, vol. 23, no. 3, pp. 1441-1448, Sept. 2008. [Baidu Scholar]

D. J. Sailor and J. R. Muñoz, “Sensitivity of electricity and natural gas consumption to climate in the U.S.A. – methodology and results for eight states,” Energy, vol. 22, no. 10, pp. 987-998, Oct. 1997. [Baidu Scholar]

T. Ahmed, K. M. Muttaqi, and A. P. Agalgaonkar, “Climate change impacts on electricity demand in the state of New South Wales, Australia,” Applied Energy, vol. 98, pp. 376-383, Oct. 2012. [Baidu Scholar]

S. M. Howden and S. Crimp. (2001, Aug.). Effect of climate and climate change on electricity demand in Australia. [Online]. Available: https://www.mssanz.org.au/MODSIM01/Vol%202/Howden.pdf [Baidu Scholar]

C. G. Hing and L. T. Tjing, “The impact of climate change on New Zealand's electricity demand,” in Proceedings of IEEE International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Singapore, Jun. 2010, pp. 1-9. [Baidu Scholar]

D. J. Sailor and A. A. Pavlova, “Air conditioning market saturation and long-term response of residential cooling energy demand to climate change,” Energy, vol. 28, no. 9, pp. 941-951, Jul. 2003. [Baidu Scholar]

T. Wilson, L. Williams, J. Smith et al., “Global climate change and California: potential implications for ecosystem health and the economy,” California Energy Commission, California, Consultant Report 500-03-058CF, Aug. 2003. [Baidu Scholar]

J. A. Dirks, W. J. Gorrissen, J. H. Hathaway et al., “Impacts of climate change on energy consumption and peak demand in buildings: a detailed regional approach,” Energy, vol. 79, pp. 20-32, Jan. 2015. [Baidu Scholar]

Y. Feng and S. M. Ryan, “Day-ahead hourly electricity load modeling by functional regression,” Applied Energy, vol. 170, pp. 455-465, Mar. 2016. [Baidu Scholar]

K. M. Powell, A. Sriprasad, W. J. Cole et al., “Heating, cooling, and electrical load forecasting for a large-scale district energy system,” Energy, vol. 74, pp. 877-885, Sept. 2014. [Baidu Scholar]

J. D. Black and W. L. W. Henson, “Hierarchical load hindcasting using reanalysis weather,” IEEE Transactions on Smart Grid, vol. 5, no. 1, pp. 447-455, Jan. 2014. [Baidu Scholar]

Y. He, R. Liu, H. Li et al., “Short-term power load probability density forecasting method using kernel-based support vector quantile regression and copula theory,” Applied Energy, vol. 185, pp. 254-266, Jan. 2017. [Baidu Scholar]

E. Paparoditis and T. Sapatinas, “Short-term load forecasting: the similar shape functional time-series predictor,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 3818-3825, Jul. 2013. [Baidu Scholar]

C. Tong, J. Li, C. Lang et al., “An efficient deep model for day-ahead electricity load forecasting with stacked denoising auto-encoders,” Journal of Parallel and Distributed Computing, vol. 117, pp. 267-273, Jul. 2018. [Baidu Scholar]

M. Khodayar, J. Wang, and M. Manthouri, “Interval deep generative neural network for wind speed forecasting,” IEEE Transactions on Smart Grid, vol. 10, no. 4, pp. 3974-3989, Jul. 2019. [Baidu Scholar]

Y. He, J. Deng, and H. Li, “Short-term power load forecasting with deep belief network and copula models,” in Proceedings of 9th International Conference on Intelligent Human-Machine Systems and Cybernetics (IHMSC), Hangzhou, China, Aug. 2017, pp. 1-5. [Baidu Scholar]

B. Yildiz, J. I. Bilbao, and A. B. Sproul, “A review and analysis of regression and machine learning models on commercial building electricity load forecasting,” Renewable and Sustainable Energy Reviews, vol. 73, pp. 1104-1122, Jun. 2017. [Baidu Scholar]

D. H. Vu, K. M. Muttaqi, and A. P. Agalgaonkar, “Short-term load forecasting using regression based moving windows with adjustable window-sizes,” in Proceedings of IEEE Industry Applications Society Annual Meeting, Vancouver, Canada, Oct. 2014, pp. 1-10. [Baidu Scholar]

S. Jovanović, S. Savić, M. Bojić et al., “The impact of the mean daily air temperature change on electricity consumption,” Energy, vol. 88, pp. 604-609, Aug. 2015. [Baidu Scholar]

T. Ahmed, D. H. Vu, K. M. Muttaqi et al., “Load forecasting under changing climatic conditions for the city of Sydney, Australia,” Energy, vol. 142, pp. 911-919, Oct. 2018. [Baidu Scholar]

Australian Energy Market Operator. (2020, Dec.). Electricity price and demand. [Online]. Available: http://www.aemo.com.au [Baidu Scholar]

Bureau of Meteorology. (2021, Jan.). Warmings current. [Online]. Available: http://www.bom.gov.au [Baidu Scholar]

Y. Hong, H. Chang, and C. S. Chiu, “Hour-ahead wind power and speed forecasting using simultaneous perturbation stochastic approximation (SPSA) algorithm and neural network with fuzzy inputs,” Energy, vol. 35, no. 9, pp. 3870-3876, Sept. 2010. [Baidu Scholar]

P. M. Berthouex and L. C. Brown, Statistics for Environmental Engineers. Boca Raton: Lewis Publishers, 2002. [Baidu Scholar]

W. W. S. Wei, Time Series Analysis – Univariate and Multivariate Methods. San Francisco: Pearson Addison Wesley, 2006. [Baidu Scholar]

D. C. Montgomery and G. C. Runger, Applied Statistics and Probability for Engineers. Hoboken: John Wiley & Sons, Inc, 2003. [Baidu Scholar]

Address:No.19 Chengxin Avenue, Jiangning District, Nanjing 211106, China

E-mail: mpce@alljournals.cn

Tel:86-25-81093060

Fax:86-25-81093040

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher