A residual load modeling approach for household short-term load forecasting application

Energy & Buildings 187 (2019) 132–143

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Energy & Buildings journal homepage: www.elsevier.com/locate/enbuild

A residual load modeling approach for household short-term load forecasting application Fatima Amara a,∗, Kodjo Agbossou a, Yves Dubé b, Sousso Kelouwani b, Alben Cardenas a, Sayed Saeed Hosseini a a b

Department of Electrical and Computer Engineering, Université du Québec à Trois-Rivières, Trois-Rivières, Québec G9A 5H7, Canada Department of Mechanical Engineering, Université du Québec à Trois-Rivières, Trois-Rivières, Québec G9A 5H7, Canada

a r t i c l e

i n f o

Article history: Received 21 March 2018 Revised 8 December 2018 Accepted 9 January 2019 Available online 22 January 2019 Keywords: Electric load forecasting Residual demand modeling Daily electricity usage Kernel density Occupant behavior modeling

a b s t r a c t The household residual component of total power consumption can be considered as a portion of load demand describing the non-temperature-related factors. This component can be decomposed to irregular and predictable energy demands. The predictable component of the residual load include consumptions which are likely to have a periodic behavior. The modeling of this periodic part of the residual load can help to enhance the overall forecasting accuracy. This paper intends to model the periodic part of the residual component by capturing the behavioral patterns of overestimated and underestimated residuals as the divisions of the main residual component. Accordingly, in order to achieve its ambition, this work proposes an Adaptive Circular Conditional Expectation (ACCE) method on the basis of circular analysis to define the sub-residuals operation schedules. Consequently, an adaptive Linear Model (LM) procedure is employed to predict the residual component demand using the results of the ACCE process at each time window. Subsequently, the predicted residual is utilized to adaptively improve the performance of total electricity demand forecasting. The accuracy of the forecasting results is evaluated using Normalized Mean Absolute Error (NMAE). As a result, the proposed approach of the periodic residual demand modeling in a daily horizon leads to a promising accuracy increase of 23%. Furthermore, the proposed residual modeling method, in combination with the temperature-related component forecasting, can increase the total power consumption prediction performance by 7%. The efficacy of the proposed approach is examined via numerical analysis of real data. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Short-Term Load Forecasting (STLF) enables the power grid industry to manage many frequent operation decisions [1,2]. STLF is an active area of research, particularly in the residential sector where high electricity consumption necessitates demand-side management strategies [3]. The STLF role is improved with enhanced load monitoring approaches that utilize load disaggregation methods [4]. For instance, in Canada where this study is conducted, the yearly average of residential buildings energy consumption is around 27 MWh [5]. This amount of energy demand expresses a great potential in employing STLF services for building energy management scenarios [6]. Household electricity consumption forecast is affected by multiple stochastic factors, including weather conditions and occupant behavior [7–9]. The contribution of these factors to the total ∗

Corresponding author. E-mail address: [email protected] (F. Amara).

https://doi.org/10.1016/j.enbuild.2019.01.009 0378-7788/© 2019 Elsevier B.V. All rights reserved.

electricity demand is explored through a correlation analysis [10]. However, the forecasting practice through analyzing the effective variables such as occupant activities is a burdensome task due to their extreme volatility. Santin et al. [11] demonstrated that occupants activities and interaction with various building components significantly affect the accuracy of energy consumption predictions, even if other effective factors such as weather conditions are well-modeled. Warren and Parkins [12] explained a relationship between occupant behavior and weather conditions. They indicated that the primary factor motivating occupants to open windows in the winter is the variation of the indoor and outdoor temperature difference considering the comfort level. Likewise, Wang and Rija [13,14] investigated the impact of different types of weather conditions on occupant energy behavior. Generally, occupant activities in building simulation tools are conventionally represented in terms of static schedules [15]. Such simplification does not properly explain the dynamic of occupant behavior on building energy consumption [16,17]. However, the

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Nomenclature k d dm ek , ε k fk (y|xk ) h( x k ) n rk R+ R T tk xk w,  yk

Discrete time Time grid schedule Middle of time period Prediction error at time k of LM and AR models Conditional density function at time k Temperature-related component (W) Number of observations Residual component (W) Positive real numbers Inverse autocorrelation matrix Time period Real time schedule Outside temperature (C) Coefficient vector of LM and AR models Aggregated power(W)

Greek symbols α Learning rate K Kernel function μ Forgetting factor ν Bandwidth of the Kernel τ Prediction horizon Acronyms STLF Short-Term Load Forecasting ACCE Adaptive Circular Conditional Expectation LM Linear Model ACDE Adaptive Conditional Density Estimation RLS Recursive Least-Squares KDE Kernel Density Estimation NMAE Normalized Mean Absolute Error AR Autoregressive Model ARX Auto-regressive with eXogenous input

dynamic interactions between building and occupants can be efficiently characterized using different statistical models of building automation control systems [18]. Yu and Fung [19] utilized a clustering analysis to identify the effect of occupant behavior on building energy consumption. They expressed that their method facilitated the evaluation of building energy-saving potential through providing a multifaceted insights into end-use patterns of building energy associated with the occupant behavior. Jang and Kang [20] developed a probabilistic model of occupant behavior based on Gaussian Process classification to control heating and electricity in apartment buildings. Their method aimed to reduce uncertainties related to occupant activities, and provide a broader application for buildings with similar conditions. Rafsanjani [21] developed a non-intrusive occupant load monitoring (NIOLM) approach to profile individual occupants energy-use behaviors at their entry and departure events. The NIOLM approach correlated occupancysensing data captured from existing Wi-Fi networks with aggregated building energy-monitoring data to disaggregate building energy loads to the level of individual occupants. Wang et al. [22] proposed an approach to simulate building occupancy using Markov chains where each occupant was associated with a homogenous Markov matrix to infer the stochastic movement process. Beside, there are other factors that can influence the energy consumption forecast however, they have not been much into consideration. In this regard, Ekici [23] exploited the artificial neural networks to predict the energy consumption using the information of building orientation and thermo-physical properties of building envelope materials.

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In fact, highly stochastic nature of occupant behavior for which there is no appropriate technical methods to correctly measure it, signifies the utilization of other effective factors specifically, in the context of non-intrusive approaches. Correspondingly, the evaluation of weather-related factors can be regarded as a useful primary analysis to reduce the modeling complexity particularly, in Nordic countries like Canada, up to 60% of the total energy consumption is related to the weather. The seasonal variations of the outside temperature are well understood, and the weather-induced departures of the actual day-to-day temperatures from these averages can be considered a stochastic process. The effect on electricity consumption of occupant activities is, however, another matter, as some such as time of day or day of the week may be foreseeable, whereas others may occur much more haphazardly. The total energy consumption of any day is thus represented by a sum of two terms, one of which is explicitly a function of the outside temperature, and the other (the residual) being that of the remainder. Due to the notable contribution of the periodic part of residual component in load forecasting, its modeling can assist to enhance STLF in the residential sector. In the literature, the residual component modeling has been investigated using different methods. Hobby and Tucci [24,25] utilized a least square fitting method to extract weather-related components in order to provide electricity forecast. They used several factors such as temperature, humidity, and atmospheric opacity in their proposal. Furthermore, they examined the hourly residual load using spectral analysis [26]. In fact, they have exploited a massive amount of data, which can increase the computational time as well as the cost of the data acquisition system. Hyndman [27,28] studied the residual load by capturing the trend and cyclic behaviors of the residual data using a parametric regression based on different time series methods such as AR model to analyze the autocorrelation patterns. The stochastic nature of residual analysis signifies the statistical approaches that rely on utilizing the non-deterministic mechanisms [29]. Such techniques can be enhanced by developing adaptive processes capable of on-line tracking of residual fluctuations. This paper proposes an adaptive non-parametric approach to extract the behavior of the periodic part of residual load with no prior information about weather conditions and occupant activities. The residual component modeling is executed using underestimated and overestimated residual profiles captured from the main residual component division. Accordingly, the periodic behavior of underestimated and overestimated residuals is investigated separately through a circular density analysis to determine each component’s operation schedule. The circular analysis utilizes an Adaptive Circular Conditional Expectation (ACCE) approach on the basis of Kernel Density Estimation (KDE) [30,31] to estimate the periodic patterns in terms of daily/weekly operation schedules. Consequently, a Linear Model (LM) is employed to predict the residual component using the results of the ACCE process in capturing the related schedules. Subsequently, the results of the residual component analysis are added to the temperature-related component decomposition forecast [32] to evaluate the forecasting accuracy of total household electricity demand in a daily horizon. Moreover, the efficacy of the proposed approach is examined through a comparison study with adaptive AR and ARX models as other effective approaches to residual load forecasting. Given the above, the significant contributions to the proposed approach to residual component modeling can be itemized as below:

1. Providing a mechanism capable of capturing the operation schedule of the periodically-related demand of total residual load by dividing it into sub-residual components consisting of underestimated and overestimated residual profiles.

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Fig. 1. Block diagram of temperature-related and residual components decomposition [32].

2. Developing an adaptive procedure capable of realizing a learning procedure by tracking the behavior changes of underestimated and overestimated residual profiles with no prior knowledge as to the nature of the household data. The remainder of the paper is organized as follows: the problem formulation is explained in section II. The residual component modeling methodology to capture the periodic part of the overestimated and underestimated residual profiles is described in section III. The process of residual load forecast using the conditional expectation results is also developed in this section. Section IV discusses the results of the real data analysis using the proposed approach. This is followed by the conclusions in section V.

2. Electricity demand decomposition In this section, the description of the problem as the accurate modeling of the residual component is provided in order to improve the performance of the STLF. The real data utilized for this process consists of hourly electricity power consumption and outside temperature information for a single bungalow-type house located in Montreal which has meteorological conditions typical of Quebec. The total electricity demand is considered as the sum of the temperature-related and residual components as expressed by (1):

yk = h ( xk ) + rk

(1)

where at discrete time k, yk is the total power consumption, xk represents the outside temperature, and h(xk ) is the temperaturerelated component [32]. rk is the residual load, which depends mainly on the variations of the number and activities of the occupants, the type of appliances usage, and environmental conditions such as wind speed and sun intensity [33,34]. Indeed, our ultimate goal is to design a structure capable of on-line modeling and forecasting of total electricity demand yk+τ for each time horizon τ . Since we consider the residual load as the difference between the total power consumption and the extracted temperaturerelated component, it is necessary to address the approach employed to capture h(xk ). In fact, the forecasting values of the temperature-related component h(xk+τ ) are estimated by investigating the outside temperature fluctuations through a correlation analysis as is shown in Fig. 1. As a result, a successful aspect of temperature-related component analysis can result in an exact estimation of residual load which in turn can notably assist an enhanced modeling practice. The temperature-related component study utilized in this paper is based on a statistical approach proposed by the authors in [32].

2.1. Temperature-related component analysis The modeling strategy is to decompose the electricity demand into temperature-related and residual components using the temperature information as the only source of data. Ultimately, the total STLF is obtained by the sum of the two components. In [32] we proposed an approach based on conditional expectation of power consumptions to define the temperature-related component of power consumption as defined by (2).

h ( xk ) = E ( yk |xk ) =



y f k ( y|x )

(2)

y

where f is the conditional distribution estimated at each time k [32]. Fig. 1 shows the mechanism that we have used to extract the temperature-related component h(xk ) through a correlation analysis. Indeed, the approach of the temperature-related component investigation is discussed due to its importance as a prerequisite necessity regarding the residual component modeling. In fact, a successful analysis of residual component is influenced by the effectiveness of the temperature-related component modeling since the residual is defined as the difference between total power consumption and the estimated temperature-related values. 2.2. Residual component analysis The residual load is computed as the difference between the measured total power yk and the estimated temperature-related h(xk ) at each discrete time k expressed by (1). The residual load modeling starts with its horizontal division to underestimated rk+ and overestimated rk− residual components given by (3), (see Fig. 2).

rk = rk+ + rk−

(3)

In fact, the underestimated and overestimated residual components as a consequence of the temperature-related component h(xk ) variations at each discrete time k can be explained by (4):



rk+ =

rk , if yk > h(xk ) , 0, otherwise



rk− =

rk , if yk < h(xk ) 0, otherwise

(4)

The underestimated residual component accounts for household heat loss related to the appliances operation and the building infiltration/exfiltration system. On other hand, the overestimated residual is related to the building heat gain generated primarily by occupant activities and sunlight [35]. As discussed, the operation schedule of the underestimated and overestimated residuals are estimated within an appropriate window in which these components are likely to have a periodic behavior. Fig. 3 illustrates the measured total power consumption and the computed sub-residual components during one year. The asymmetrical behavior of underestimated and overestimated residual

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Fig. 2. Block diagram of residual component modeling approach.

Fig. 3. Yearly electricity consumption, temperature-related and residual components with detailed seasonal description during one day of winter (A) and summer (B).

profiles demonstrates different behavioral patterns in terms of high level spikes of short duration in the former and an inverse behavior in the latter. Therefore, our residual component modeling strategy aims to capture the periodic behavior of the underestimated and overestimated residual time series through two separate analyses. Accordingly, a circular distribution estimation is applied to a specific period of time where these components indicate a likely periodic behavior in order to define their operation schedules. Subsequently, due to STLF purposes, the daily and weekly periodicity of underestimated and overestimated residual components are examined using their conditional expectations over the related time period given hourly data samples tk defined by (5).

g(tk ) = E(rk |tk ) =

 r

r fk (r |t )

(5)

3. Residual component modeling The adaptive mechanism can enable an effective investigation of periodic behavior of the underestimated and overestimated residual components. Accordingly, the adaptive procedure can provide the estimate operation schedule of rk+ and rk− residual profiles of related data from daily/weekly periods. In these periods, the underlying stochastic phenomena of the residual component can include circular observations, which subsequently can cause a likely periodic behavior in the residual load. Therefore, an examination of underestimated and overestimated operation schedules resulting from a periodic behavior analysis allows to explain the amount of electrical demand corresponding to each factor during a specific period. Afterwards, the operation schedules defined using the examined conditional expectations of ACCE process are exploited to

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estimate the residual components on the basis of a linear modeling method. 3.1. Adaptive circular conditional expectation method The divided residual components rk+ and rk− vary due to the changes in electric appliances usage and climate conditions. According to (5), the conditional expectation g(tk ) in the real time tk of time calendar data belong to the time of day/week vector  ={0, 1, 2, .., 23}, is defined using the conditional distribution function fk (r|t). This function requires the computation of the marginal distribution fk (t) and the joint distribution fk (r, t) functions. However, in order to simplify the calculation complexity, g(tk ) is estimated directly by the ACCE method to avoid the computation of fk (r|t). Accordingly, the behavior patterns of rk+ and rk− are adaptively examined to describe their daily/weekly operation schedules as a periodic function g : R+ → R+ with period T, described by (6):

g(tk ) = g(tk + T )

(6)

Subsequently, the expected power values of underestimated and overestimated residuals are defined using the operational schedules. The conditional expectation g(tk ) can be estimated using a univariate Gaussian-based kernel estimator expressed by (7), starting at the middle of the time periods, i.e. dm = T2 which is 12 pm for both daily and weekly operation schedules to ensure a symmetric kernel distribution [36]. The kernel function K is constructed over an hourly discrete time grid d. Subsequently, the kernel estimator of new residual component arrivals is examined by a time shift to K (d − tk ) in order to keep the symmetric distribution over the analyzed points.



K (d ) = exp −

2

1 ( d − dm ) 2 ν2



(7)

where ν is the bandwidth value of the kernel function, which has been set to 1.0 hour for both rk+ and rk− . One-hour bandwidth gives a better estimation compared to other hourly intervals up to 24 hours. However, this selected bandwidth shall not apply to all the houses with different power consumption. It should opt carefully, considering the data sample time. As a result, the adaptive procedure revises the residual expectation values using (8):

gk (d ) =gk−1 (d ) + α (rk − gk−1 (d ) )K (d − tk )

(8)

where g is the expected value of the residual at time instant k, and α is the learning rate set to 0.01 in order to ensure the model’s convergence [37]. The yearly conditional expectation distribution captured by the adaptive ACCE mechanism for both rk+ and rk− is shown in Fig. 4. The red-brown regions illustrated in Fig. 4(a) and (b) are the expected energy demand with high power consumption for underestimated residual and high gain amount for overestimated residual. The increase in the expected values of rk+ on weekends can be due to the fact that the household appliances which influence this factor (such as dryer and oven) are more probable to be utilized at this time. On the other hand, the high gain regions for the expected values of rk− during the summer and autumn periods demonstrate the high household heat gain from phenomena like occupants and sun. Fig. 4(c) depicts the residual operation schedule for one week. It can be seen that the value and occurrence time of spikes in rk+ during morning and evening and rk− during night are different. In addition, a likely periodic behavior can be observed in both rk+ and rk− consumption trends separately, which strengthen the concept of the proposed adaptive circular analysis. Accordingly of daily operation schedule, the rk+ and rk− residual components captured using the circular distribution analysis at the

end of the process in winter and during the procedure in summer are illustrated in Fig. 5(a) and (b), respectively. The rk+ and rk− operation schedules extracted through the adaptive procedure ACCE can provide a practical interpretation of household heat loss and gain mechanisms. It can be expected that the rk− is likely to have a uniform distribution in winter due to occupant activities of selecting almost constant day/night set-point temperatures for the entire cold season. Subsequently, the increase in expected power values at night is because of the thermostats’ regulation to a lower set-point temperature. On the other side, the growth in rk− during day in summer time is mainly due to sun heat gain. 3.2. Linear regression model The daily/weekly underestimated and overestimated residual operation schedules captured from the yearly analysis are entered into a linear regression model (LM). The LM which is based on RLS filter, attempts to adjust the model coefficients wk using the prediction error ek defined as the difference between the original residual rk and the estimated residual rˆk of one year analysis. The RLS inputs defined in (9) are the expected underestimated and overestimated residual values estimated using the operation schedules which are added to compute the total reconstructed residual component. Concurrently, the coefficients of the filter are adaptively updated using the conditional expectation of arrival values at each time interval [38,39].



uk = E(r + |tk ), E(r − |tk )

T

(9)

The RLS filter output is given by (10):

rˆk = wk uk

(10)

where uk is the input vector for computing the coefficient vector wk = [w1k , w2k ]T related to the E(r + |tk ), and E(r − |tk , respectively, using (11).

wk+1 = wk + Gk ek

(11)

That Gk is the gain vector expressed by (12):

Gk =

μ−1 Rk−1 uk T 1 + μ−1 uk Rk−1 uTk

(12)

where Rk is the inverse correlation matrix [38], given by (13):

Rk = μ−1 [Rk−1 − Gk uk Rk−1 ]

(13)

The forgetting factor value of RLS algorithm is μ = 0.99, in accord with [38]. As a result, the residual component rˆk+τ on the time horizon τ can be estimated using the daily/weekly operation schedule information tk+τ and the trained linear model, as illustrated in Fig. 6(b). The schematic diagram depicts the structure of the residual component modeling and forecasting using the combined approach of ACCE/LM. 4. Case comparison and discussion 4.1. Comparative methods A comparison study is provided using the adaptive AR and ARX models which consider no division process of the residual component. The adaptive AR model presented in [40] is used for forecasting the residual component based on (1). By definition the output of the AR model is defined using (14):

rˆk+τ =

p 

θ j rk+τ − j + εk+τ

(14)

j=1



T

where, ε k is the prediction error, and k = θ1k , θ2k , . . . , θ pk is the coefficient vector. Accordingly, we have utilized an input vector

F. Amara, K. Agbossou and Y. Dubé et al. / Energy & Buildings 187 (2019) 132–143

(a)

137

(b)

(c) Fig. 4. Daily/weekly conditional expectation distribution in terms of density variations during one year of analysis for underestimated (a) and overestimated (b) residuals absolute values, (c) yearly analysis results of the last week of December.

zk of six residual data for the AR model expressed by (15):



zk = rk−1 , rk−2 , . . . , rk−p

T

(15)

In addition, an ARX model which utilizes a daily/weekly operation schedule tk as an explanatory variable is considered according to [41,42]. The output of this model is defined by (16):

rˆk =

p  j=1

θ j rk− j +

b 

(18) and (19). It is noted that the estimated and desired values of the analysis process are the residual component and the total power forecasting values at time step k.

NMAE =

n ˆk | k=1 |qk − q n k=1 |qk |

1 |qk − qˆk | N

(17)

N

ϕitk−i + ck

(16)

i=1

where, ϕ i represents the coefficients of tk and c is the error. The maximum number of residual data and tk time lags are supposed to be p = 6 and b = 6, respectively. The prediction horizon τ is considered to be 24 hours stepahead in order to predict rk+τ values using the models described above. The accuracy metric used to evaluate the performance of the residual component modeling and forecasting is defined using the Normalized Mean Absolute Error (NMAE), Mean Absolute Error (MAE) and Correlation Coefficient (R-Square), given by (17),

MAE =

(18)

k=1

N (qˆ − q¯ k )2 R2 = kN=1 k 2 k=1 (qk − q¯ k )

(19)

where qk represents rk for the residual component and yk for the total power consumption values. 4.2. Results and discussion The discussion focuses on exploring the results of both residual component and total electricity demand forecasting in two subsections as bellow.

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Fig. 5. Daily conditional expectation distribution of underestimated and overestimated residual in kW for Friday, December 15th (a), and Monday, July 31th (b).

Fig. 6. Block diagram of the combined approach ACCE/LM for modeling and forecasting the residual component.

4.2.1. Residual forecasting results Fig. 7 compares the results between the residual component forecast using the proposed approach and two other selected models. It can be observed that ACCE/LM approach can provide the most accurate description of the original residual component compared to other models. It is evident that the ACCCE/LM approach gives a much better fit to the estimated residuals than do the two other methods. The AR model does a creditable job following the short-term peaks and valleys, and that much better than what the ARX model can. Fig. 7(A) depicts the capability of the discussed models to track power consumption peak demands, specifically in warm seasons where a considerable part of power consumption depends on occupant stochastic behavior. Since this demand which can have a periodic behavior accounts for an important part of residual component, it can be explained by applying the ACCE/LM approach. Therefore, it can be seen that this approach provides a better explanation of peak demands. Indeed, our combined approach excels other models due to its capability in providing a successful conditional expectation distribution over the residual component fore-

cast. On the other hand, the drawback of AR and ARX models can be attributed to the fact that they are parametric methods, which require a set of a fixed number of coefficients. We would like to note here that the NMAE’s values have been described in terms of accuracy percentage, i.e. (1 − NMAE ) × 100 in the related discussions of results. Fig. 8 shows the computed NMAE, MAE and R-Square of each forecasting model applied to the residual component. It is evident that there is again a notable difference in NMAE, MAE, and R-squared among the models in a way that AR and ARX models have higher error. These models indicate poor performance results compared to the ACCE/LM approach which is capable of explaining around 23% of the total residual component in all time horizons. In fact, the proposed ACCE/LM is an effective method because it targets a portion of the residual component which exposes a circular observation to capture its periodic part. 4.2.2. Electricity demand forecasting results The combined ACCE/LM approach to the residual component modeling can subsequently leas to an improved forecasting of total power consumption. Accordingly, in order to evaluate the efficacy of the approach in total power demand forecast, the temperaturerelated component of the total consumption is predicted on the basis of the method proposed by the authors in [32]. Fig. 9 shows the comparison results of the yearly total power consumption prediction among the utilized models. It can be seen that the ACDE method in forecasting the temperature-related component faces difficulties in capturing peak demands due to the fact that power consumption increases in peak periods is because of the stochastic phenomena such as occupant activity that most strongly influence residual consumption. Consequently, we combined the residual component modeling approaches, i.e. (AR, ARX, ACCE/LM) with the temperature-related component forecasting method in order to enable a total power consumption prediction procedure. Fig. 10 shows the accuracy results of the composed models as well as the temperature-related component model in total demand forecasting. It can be seen that the total models have achieved better fit to the electricity demand data which evidences

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Fig. 7. The forecasting results of residual component analysis using different methods in a daily horizon.

Fig. 8. (A): NMAE, (B): MAE and (C): R-Square comparison results of different residual forecasting models as a function of the forecast lead time.

139

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Fig. 9. Yearly total electricity demand forecasting results using different models in a daily horizon with a detailed prediction outcome of two days.

Fig. 10. Total electricity demand forecasting accuracy before and after adding the residual component modeling approaches using different methods as a function of the forecast lead time..

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Fig. 11. Total electricity demand forecasting improvement resulting from adding the temperature-related component modeling method to the residual component modeling approaches..

the capability of the residual modeling approach in explaining the non-temperature-related power consumption. Furthermore, the ACDE+ACCE/LM method provides a notably improved accuracy in forecasting during all time horizons compared to the other composite models. This approach can thus reconstruct the total power consumption in the context of a STLF mechanism and account for a satisfactory degree of electricity demand explanation, specifically in terms of the power variation tracking. In fact, the ACDE is capable of explaining approximately 68% of the total electricity demand in all time horizons [32] as shown in Fig. 10. This accuracy rate increases to 75% by adding the results of residual analysis modeling considering the NMAE metric. This considerable growth in the accuracy of total power consumption forecast is further remarkable using MAE and R-Square metrics that manifests the efficacy of ACDE+ACCE/LM technique. Fig. 11 demonstrates this forecasting improvement rate as a result of adding the residual component modeling analysis in order to clarify the effectiveness of the proposed residual modeling in enhancing the electricity demand prediction. Subsequently, it can be observed that our proposed approach can significantly improve

the total power consumption prediction by about 7% in all time horizons. Indeed, the best results of total electricity demand forecasting among other models is provided by the ACDE+AR model in the time horizon of τ = 24 hours with a 3% improvement. Accordingly, the present analysis demonstrates the efficacy of the proposed ACCE/LM method in residual component modeling with regard to a fruitful residential electricity demand forecasting using real data. During this study, the proposed approach has demonstrated a promising improvement in household electricity demand forecasting. However, it reveals some limits related to the bandwidth value of Kernel estimator. The bandwidth choice is one of the major difficulties of the proposed non-parametric approach is to find a suitable bandwidth regarding the new observations arrival in an online system. Even in the case of using a constant bandwidth, different houses require their specific bandwidth. Indeed, the bandwidth can highly influence the results of the Adaptive Circular Conditional Expectation (ACCE/LM) approach. Another drawback of the proposed modeling structure is that it faces difficulties to apply for multi-dimensional analysis and needs to be modified.

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5. Conclusion This work has proposed an approach to the modeling of residual component of household electricity demand. This approach captures the behavior of the periodic part of the residual load with circular observations which is likely to have a periodic pattern over a specific time horizon. In fact, the proposed method extracts daily/weekly operation schedules of the underestimated and overestimated residual component profiles through a circular analysis. The modeling structure utilizes an Adaptive Circular Conditional Expectation (ACCE) method which is subsequently combined with a Linear Model (LM) to construct a residual component forecasting procedure. The effectiveness of the proposed approach is demonstrated through a comparative study with other STLF methods. In addition, the total household electricity consumption forecasting is provided by adding the results of the residual modeling process and temperature-related component modeling analysis. As a result, the composed forecasting models can achieve satisfactory results of 75% in explaining the total power consumption in a dayahead forecast horizon. The applicability of the proposed modeling process is significant, since it utilizes only one source of information i.e. the outside temperature to provide the entire forecasting procedure. Acknowledgments The authors would like to thank Nilson Fernaldo Henao for his valuable suggestions to improve the quality of the paper, Jonathan Bouchard and Michael Fournier for their cooperation to provide the real data of household electricity. Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.enbuild.2019.01.009 References [1] A.K. Srivastava, A.S. Pandey, D. Singh, Short-term load forecasting methods: A review, in: 2016 International Conference on Emerging Trends in Electrical Electronics & Sustainable Energy Systems (ICETEESES), IEEE, 2016, pp. 130–138, doi:10.1109/ICETEESES.2016.7581373. [2] Y.-h. Hsiao, Household electricity demand forecast based on context information and user daily schedule analysis from meter data, IEEE Trans. Ind. Informat., vol. 11, no. 1, February 2015 11 (1) (2015) 33–43. [3] C.-N. Yu, P. Mirowski, T.K. Ho, A sparse coding approach to household electricity demand forecasting in smart grids, IEEE Trans. Smart Grid (2016) 1–11, doi:10.1109/TSG.2015.2513900. [4] N. Henao, K. Agbossou, S. Kelouwani, Y. Dube, M. Fournier, Approach in nonintrusive type i load monitoring using subtractive clustering, IEEE Trans. Smart Grid (2015), doi:10.1109/TSG.2015.2462719. 1–1 [5] Statistics Canada, L’enquête sur les ménages et l’environnement : utilisation de l’énergie, 2013, Technical Report, 2016. [6] T. Teeraratkul, D. O’Neill, S. Lall, Shape-Based approach to household electric load curve clustering and prediction, IEEE Trans. Smart Grid (2017), doi:10. 1109/TSG.2017.2683461. 1–1 [7] S. Khatoon, Ibraheem, A.K. Singh, Priti, Effects of various factors on electric load forecasting: An overview, in: 2014 6th IEEE Power India International Conference (PIICON), IEEE, 2014, pp. 1–5, doi:10.1109/POWERI.2014.7117763. [8] D. Yan, W. O’Brien, T. Hong, X. Feng, H. Burak Gunay, F. Tahmasebi, A. Mahdavi, Occupant behavior modeling for building performance simulation: current state and future challenges, Energy Build. 107 (2015) 264–278, doi:10. 1016/j.enbuild.2015.08.032. [9] D.G. Stephenson, Thermal radiation and its effect on the heating and cooling of buildings, Technical Report, National Research Council Canada. Division of Building Research, 1961. [10] F. Amara, K. Agbossou, Y. Dube, S. Kelouwani, A. Cardenas, Estimation of temperature correlation with household electricity demand for forecasting application, in: IECON 2016 - 42nd Annual Conference of the IEEE Industrial Electronics Society, IEEE, 2016, pp. 3960–3965, doi:10.1109/IECON.2016.7793935. [11] O. Guerra Santin, L. Itard, H. Visscher, The effect of occupancy and building characteristics on energy use for space and water heating in dutch residential stock, Energy Build. 41 (11) (2009) 1223–1232, doi:10.1016/j.enbuild.2009.07. 002.

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