A Probabilistic Approach to Study the Load Variations in Aggregated Residential Load Patterns Intisar A. Sajjad, Gianfranco Chicco, Roberto Napoli Energy Department Politecnico di Torino Torino, Italy
[email protected] Abstract— The demand side in a power system has key importance in the evolving context of the energy systems. Exploitation of possible flexibilities of the customer’s behavior is considered as an important option to promote demand response programmes and to achieve greater energy savings. For this purpose, the first action required is to augment availability of information about consumption patterns. The electricity consumption in a residential system is highly dependent on various types of uncertainties due to the diverse lifestyle of customers. Knowledge about the aggregated behavior of residential customers is very important for the system operator or aggregator to manage load and supply side flexibilities for economic operation of the system. In this paper, the effect of sampling time is evaluated for different residential load aggregations using probabilistic approach. A binomial probability distribution model is used to extract trends in increase or decrease in demand with respect to time evolution of a typical day. For each case study scenario, confidence intervals are calculated to assess the uncertainty and randomness in load variation trends. The findings of this study will lead towards better management of demand and supply side resources in a smart grid and especially for microgrids. Keywords— Binomial distribution; load variation pattern; aggregation; sampling time; confidence intervals, maximum likelihood estimation
NOMENCLATURE CI DR MLE
∆
, ∆ , ∆ /
th
Confidence interval. Demand response. Maximum Likelihood Estimation. Aggregation level. Total no. of measurments (samples) for sampling interval . Observation number. Power demand at time instant ∆ . Sample interval. Outcome of a Bernoulli trial. Binomial discrete random variable. Critical value of the normal distribution at significance level .
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∆
Change in load at time instant ∆ between two successive measurements for any combination of , and . Duration of sampling interval for sampling interval . Probability of binomial discrete random variable. Estimated value of , ∆ using MLE.
, ∆
∆ , ∆
, ∆
∆
, ∆
,
, ∆
,
, ∆
, ∆
Upper and lower limits of the Wilson Score Interval. Relocated mean and standard deviation for the Wilson Score Interval. Defines CI width, i.e. 100 1 %. Load pattern of a typical day. Load variation pattern of a typical day. I.
INTRODUCTION
Characterization of electric load profiles is of great relevance for power distribution system studies. Effective management, planning and operation of local generation and electrical demand are very much dependent on the aggregated behavior of customers to be supplied [1-4]. The studies about the nature of electricity consumption are gaining importance also because of the technological advancements in metering, automation and control, information and communication. The information about time evolution of electricity consumption is very useful for the aggregator or the system operator to perform network studies, initiate demand response (DR) programmes and manage load and demand side flexibilities to produce incentives for customers [5-8]. Nowadays, the study of aggregated load profiles of specific group of customers in the energy sector is being focused to analyze the possible impact of demand response, tariff differentiation and direct load control [9-14]. The electrical load patterns that represent the consumption level are affected by different type of uncertainties associated with customer’s behavior and with keeping acceptable comfort level at the customers’ premises [9, 15]. However, assessing the time evolution of such systems is a challenging task.
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Residential loads are important because they could be conceptually available all around the year and 24 hours per day for load management purposes. However, for a small aggregation level (e.g. up to 20 consumers) the demand may changes very rapidly in time, depending on the life style and on the size of the family groups [15, 16]. Yet, on a higher level of aggregation the changes in the aggregated load pattern could be relatively lower, due to the effect of diversity in the usage of the appliances by the different customers. The aggregated load pattern represents the system response seen by the system operator or aggregator. This response may be more or less flexible, i.e., adaptable to the incorporation into DR programmes. The amount of load demand flexibility has been the subject of various research activities, also considering the uncertainties associated with the load patterns [17-22]. Yet, an approach based on the characterization of the load variations for different levels of load aggregation is substantially novel. Furthermore, the definition of the sampling time of an energy measurement device (e.g., a smart meter) is very important to extract customer’s behavior. In fact, the use of different sampling times results in different data features. The relevance of sampling time has been discussed in literature for the application of load disaggregation algorithms [23]. In aggregated demand, it is important for the aggregator or the system operator to get data with appropriate sampling time that will help them to extract sufficient information about the combined customer’s behavior. This information may be used to study better utilization of DR programmes through optimal management of demand and supply side resources. Applying a probabilistic method to model the customer’s behavior has the advantage that the results are more generalized and easy to compare. Categorical data model, based on binomial probability distribution, is presented in this paper to study the load variations for aggregated residential load patterns. Change in demand is characterized using different states, i.e., demand may increase, decrease or remain constant. These states are represented as categories, and the information about the probability of occurrence of each category at specific time instant is helpful for the system operator to manage supply and demand side resources. The next sections of the paper are organized as follows. Section II describes the characteristics of the data under study and presents the mathematical representation of data collection. Section III explains the probabilistic technique applied by using the mathematical model described in Section II. Section IV explains the application of the moving window averaging technique for trend extraction from the time evolution results explained in section III. Case studies are presented in Section V. The results of these case studies are discussed in Section VI. Conclusions and future extensions for this work, based on the case study results, are presented in Section VII. II.
DATA FOR RESIDENTIAL CONSUMERS
The aggregated residential patterns used in this study have been generated for extra urban residential consumers using
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Monte Carlo Simulation, and are based on information about the family composition and lifestyle, house characterization, usage of electrical appliances inside each type of house, directly collected from the residents [16]. Since measuring devices e.g. smart meters, log data about consumption in a discrete fashion with some sampling time (∆ ), we can represent the measured data in terms of load pattern using (1) as: ∆
,
,
∆
(1)
∆
We are interested to compute load variations to see whether load is increasing or decreasing when time evolves from one instant to another. Eqs. (2) and (3) represent the load variation pattern. The time evolution of some aggregated residential load variation patterns is shown in Fig. 1. ∆
∆
∆
∆
∆
,∆
∆
,
∆
,∆
(2)
∆
(3)
∆
1,2,3, … ,
and
Our interest is to study the effect of different sampling intervals and aggregation levels for residential customers on the load variation pattern. For each set of aggregation level and sampling interval, there are observations. Taking into account different aggregation levels and sampling intervals for each aggregation level, the data can be represented in a matrix by using the expressions (4) to (5), with 1,2, , and 1,2, , : ∆ ∆
∆
∆
,∆
∆ ∆
,∆
∆
,∆
, ∆
∆ ∆
, ∆
∆
, ∆
,
∆
,
∆
,
∆
(4)
Each row of the matrix ∆ is the load variation pattern for any aggregation level and sampling time . The time evolution of the demand variation during a day, for an aggregation of 20 houses and sampling time of 15 minutes is shown in Fig. 2. Each column of (4) is a set of observations for a particular time sample and is calculated using (3). We can represent in terms of column vectors as: ∆ ∆
∆ III.
,∆
∆
, ∆
∆
,
∆
,
(5)
CATEGORICAL DATA ANALYSIS
Different methods are used in statistics to analyze data with different characteristics. Categorical data analysis is one of the statistical approaches to analyze data for data clustering, correlation analysis, system modeling etc. and has wide applications in different fields of science and technology [2428]. The British statistician Pearson worked in this field
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around 1900, then very little development was noticed until 1960’s. From 1960 up till now, a lot of work has been done for method development related to categorical data analysis [29]. For the particular problem addressed in this paper, we consider that load variations may be positive or negative. A third possibility may exist in some cases, i.e., no change in load, meaning that the possible load variations in two consecutive demand values are lower than the amplitude resolution of the meter. This occurs especially when the meter resolution is relatively poor. The load variations are dependent on sampling time and aggregation level. Categorical analysis of such data is carried out with , as independent/explanatory variables and ∆ response variable. x 10
x 10
Demand (W)
1.5
0
Each observation is just like a Bernoulli trial with only two outcomes. The total number of observations is fixed to . Let be the outcome of each Bernoulli trial at particular , ∆ time instant ∆ and it is defined as: 0
, ∆
(6)
~
,
(7)
, ∆
4
∑
4
, ∆
2
, ∆
for
, ∆
0,1,2, … ,
, ∆
(9)
, ∆
-2 , ∆
,∆
20
-1.5
100
16 12
-2
,
, ∆
,∆
,
∆ ,
(10) ,
∆
(11)
50
8 4
Time of Day
0
0
Observations
Fig. 1. Time evolution of the aggregated load variations for 150 houses and sampling interval of 15 min. 4
x 10 1
B. Maximum Likelihood Estimation For the binomial model, presented in subsection B, all the parameters in (11) are unknown. As such, needs to be estimated using some suitable estimation technique. Maximum likelihood estimation (MLE) is a method, used to estimate the parameters of a statistical model and is widely used for estimation of binomial parameters [30-33]. We can define MLE for our problem as:
0.5 , ∆
0 , ∆
arg max
: MLE of
(12)
, ∆
, ∆
, ∆
-0.5 , ∆
, ∆
(13)
-1 0
10 12 14 16 18 20 22 24 Time of Day Fig. 2. Box plot of the demand variations for aggregation of 20 houses with 15 min. sampling time.
2
4
6
8
A. Binomial Representation There are three basic probability distributions which are used for categorical data analysis, i.e. Binomial, Poisson and Multinomial distributions. The binomial distribution is the special case of multinomial distribution with only two categories. Our problem has been modeled using the binomial distribution with two response variables: 1. Increase in demand.
th
(8)
Using (5) to (7), we can represent the set of random variables and its probability as:
0
-4 24
Demand Variation (W)
∆
For a particular set of aggregation level ( ) and sampling time ( ), let be a binomial discrete random variable, , ∆ defined as:
-0.5 -1
1, 0,
, ∆
, ∆
2
0.5
No increase in demand (including possible cases with no change in demand).
4
2.5
1
2.
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is a likelihood function of the The term , ∆ binomial parameter. According to the definition of binomial probability mass function, we can write (13) as: , ∆
, ∆
, ∆
, ∆
1
, ∆
, ∆
(14) After solving (12) using (13) and (14), the result of MLE becomes:
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, ∆
(15)
The term , ∆ is computable because all the parameters are known, and also it is an unbiased estimator because the expected value is and the variance , ∆ , ∆ , ∆
, ∆
. Using the results of MLE in (15) we can rewrite (10) by replacing (11) as: is
, ∆
,∆
, ∆
,
∆
,
Aggregation ( ) 1 2 3 4
DESCRIPTION OF DIFFERENT SAMPLING TIMES AND AGGREGATION LEVELS Description 20 houses 50 houses 75 houses 150 houses
Sampling Time ( ) 1 2 3 4
Description 5 minutes 15 minutes 30 minutes 60 minutes
C.1. Wilson Score Interval There are many established methods in literature to calculate confidence intervals [34]. The most simple and basic method is normal approximation using central limit theorem [35, 36]. This approximation fails when the trial entries are too low or the , ∆ is very close to 0 or 1. These bottlenecks were addressed by Bidwell Wilson who developed the Wilson score interval in 1927. For this method, the actual coverage probability of confidence interval is approximately equal to the nominal one, even for small no. of trials or , ∆ closer or equal to 0,1 [37]. This method was used in many research publications and has advantages over the other methods, e.g. good average coverage probability, less average expected length and smaller mean absolute error [34, 35, 38]. We have used the modified version of Wilson Score Interval method as described in [35] and the interval limits for our problem are calculated using (17) to (19).
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,
/
, ∆
, ∆
·
, ∆
, ∆
.
/
, ∆
, ∆
(17)
, ∆ /
1
, ∆
/
(18) 1
/
(19)
By using the results of Eq. (17) to (19), we can rewrite (16) as follows:
(16)
C. Confidence Interval for Binomial Proportions is directly related to and is The entry , ∆ calculated based on the outcomes of each trial, , ∆ . The probability of success is the same for each trial, and the trials are statistically independent of each other. If in one experiment we have , ∆ equals to 0.6, it may happen that in a second experiment for the same environment this may be 0.61. We cannot predict the binomial parameters with 100% accuracy, because the calculations are based on the population sample but not on the whole population. That’s why different methods are being formulated to find the confidence intervals (CIs) for binomial parameters. The CIs are very informative, because they indicate the level of uncertainty or randomness of the load increase or decrease. If in a given time period one scenario has lower CI in comparison with another, it means that the former scenario has a more regular trend about increasing or decreasing the load in that time period. TABLE I.
, ∆
, ∆
,∆
,
(20)
∆
,∆
, ∆
,
∆
(21)
,∆
, ∆
,
∆
(22)
The length of CI will simply be the difference between (21) and (22). (23) MOVING WINDOW AVERAGE
IV.
For small sampling time, data as well as the calculated responses in terms of probability are more fluctuating. Moving window averaging techniques are used to smooth out short term fluctuations and give information about long term trends. This technique has the disadvantage that we will lose some information at the start and end of time series or sequence, and the advantage is that we will get more comparable time response. Some of its applications are in signal processing, pattern recognition, power system protection [39, 40]. For the calculations presented in this paper, the simple window average method is used with window size equal to 1 hour and window step equal to the sampling time. Fig. 3 shows the application of the moving window average for with 1 and 1. 1 Moving Window Average Actual Results
0.8 Probability
, ∆
0.6 0.4 0.2 0 0
2
4
6
8
10 12 14 16 18 Time of Day
20 22
24
Fig. 3. Comparison of Moving Window Average and actual results for with 1 and 1.
V.
CASE STUDIES
For case studies we have used 4 aggregation levels with aggregation of 20, 50, 75 and 150 houses respectively.
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(a)
(e) 1
0.2
Probability
0.6
1 2 3 4
Length of CI
a= a= a= a=
0.8
0.4 0.2 0 0
2
4
6
8
10 12 14 Time of Day
16
18
20
22
0.6
1 2 3 4
Length of CI
Probability
0
2
4
6
8
10 12 14 Time of Day
16
18
20
1 2 3 4
22
24
0.2 a= a= a= a=
0.8
0.4 0.2 0 0
2
4
6
8
10 12 14 Time of Day
16
18
20
22
0.15 0.1 a= a= a= a=
0.05 0
24
(c)
0
2
4
6
8
10 12 14 Time of Day
16
18
20
1 2 3 4
22
24
(g) 1
0.2
0.6
1 2 3 4
Length of CI
a= a= a= a=
0.8 Probability
a= a= a= a=
0.05
(f) 1
0.4 0.2 0 0
2
4
6
8
10 12 14 Time of Day
16
18
20
22
0.15 0.1 a= a= a= a=
0.05 0
24
(d)
0
2
4
6
8
10 12 14 Time of Day
16
18
20
1 2 3 4
22
24
(h) 1
0.2
0.6
1 2 3 4
Length of CI
a= a= a= a=
0.8 Probability
0.1
0
24
(b)
0.15
0.4 0.2 0 0
2
4
6
8
10 12 14 Time of Day
16
Fig. 4. Comparison of for 1,2,3,4 with (a) 2 (g) 3 (h) 4.
18
1 (b)
20
22
24
2 (c)
Each aggregation has a typical daily load pattern with 4 sampling intervals, 5, 15, 30 and 60 minutes respectively. For each set, we have O = 100 observations. We can summarize this data as: 1,2,3,4 , 1,2,3,4 , 1,2,3, … ,100 . Table I provides the description for each value of and . Load variations for all combinations of and are calculated using (2) and (3) and results of load variations are stored in form of matrix as mentioned in (4) and (5). Numerical results are evaluated and expressed in terms of categorical random variables using (6) to (8). Then MLE is applied on the these results to calculate the estimated probability of increase in demand using (15) and then results are stored in the form of (16). CIs for each combination of are calculated using (17) and the final results are organized in matrices as defined in (20) to (23). VI.
RESULTS AND DISCUSSION
Fig. 4 summarizes the results for all combinations of and . Fig. 5 presents some selected events for the time
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3 (d)
0.15 0.1 a= a= a= a=
0.05 0
0
2
4
6
4 and comparison of
8
10 12 14 Time of Day
for
16
18
20
1,2,3,4 with (e)
22
1 2 3 4
24
1 (f)
evolution of load variations in Fig. 4. A general trend for change in demand can be seen in Fig. 4a. The change in electrical demand is almost constant during night from 2 to 5 a.m. since , ∆ is almost equal to 0.5. Since the probability of load decrease is simply 1 , ∆ , this means that positive and negative variations are almost equal. Another observation can also be made during the time slot when , ∆ is about 0.5 that the effect of aggregation level is very random. This effect can be seen in Fig. 5b with more detail. From Fig. 5, it can be seen that with aggregation of 20 houses ( 1), the behavior of sampling time 1 and 2 is almost the 3,4 respectively. same and , ∆ decreased with The aggregation level a = 2 shows that with increase in the probability for increase in demand decreases. For 3,4, the effect of 2,3 is unchanged, but other two combinations show decrease in , ∆ . Starting from 5 am, an increase in demand goes up until a quarter past 7 am. Then, a sharp decrease can be noted, and after 8 am the decrease in load
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variations is more prominent because , ∆ is less than 0.5. Another positive load variation trend occurs between 11 am and 2 pm. The value of , ∆ is small compared with the morning peak. Another trend for demand increase can be found from 5 pm and 9 pm. A random behavior of , ∆ occurs between 9 pm and 10 pm. A typical trend of load decrease load can be noticed from 10 pm until midnight. The effect of aggregation is prominent during the time slots when the load variations follow a trend of decrease or increase in demand. This effect is demonstrated in Fig. 4 at 7 am. Parameter , ∆ increases as we go on with higher sampling times. The value of 4, which means that , ∆ is 1.0 at there is zero probability to observe decrease in demand for given observations ( ). The confidence intervals are marked with vertical bars to show the population-based CIs. If we are we are actually getting more observing increase in , ∆ belief that the load variations are following a particular trend. This can be interpreted as we are losing the dynamics of individual customers in terms of demand flexibility. From Fig. 4a to Fig. 4d, it can also be observed that gives meaningful information up to 2. We have very less information about any flexibility information with sampling time 3 and 4.
Probability
1 0.9 0.8 0.7 0.6
s=1
s=2
s=3
s=4
0.5 20
50 75 Aggregation level (No. of Houses) (a) 7 a.m.
150
0.8 Probability
s=1
s=2
s=3
s=4
0.6 0.4 0.2 0
20
50 75 150 Aggregation level (No. of Houses) (b) 4 a.m. for all combinations of Fig. 5. Comparison of and levels of load aggregation and sampling time .
From Fig. 4e, it can be seen that most of the time the value is very close it its upper limits. This of the length of indicates that is almost equal to 0.5. So in terms of , ∆ load variations we can say that most of the time there is almost an equal probability of increase or decrease in demand. This equal probability can be due to two reasons. First, in the time is very close to 0.5 and the aggregated slots when , ∆ demand is very low (e.g., from 2 a.m. to 5 a.m.), very small numbers of events related to change in demand are noted. Secondly, if the demand is reasonably high and , ∆ is very
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close to 0.5, this information indicates that the behavior of the individual customers is very random in that time slot. This randomness means that customers may change their behavior if they will be properly incentivized through DR programs. This information is very helpful to an operator or aggregator for economic operation of system by managing supply and demand side flexibilities. Also the behavior of all aggregation levels is same except the time slots when the demand follows a specific trend. This effect is prominent when we go with higher sampling times. Fig. 4f provides clearer picture about the trends during morning and evening peaks and then decrease in demand during night after 10 p.m. The activities during the evening peak are more random in nature compared with the morning peak. This is reflected by for the evening the fact that the maximum value of , ∆ peak is lower than the one in the morning peak. So we can conclude from this observation that the load during the evening peak is more flexible in nature as compared with the morning peak. VII. CONCLUSIONS The methodology presented in this paper provides meaningful information about the effect of sampling time on aggregated load variation patterns. With the increase in sampling time a trend to lose the load variations characterizing the behaviour of the single customers is noticed, and the time evolution of the aggregated load tends to become more and more similar in shape. From the results obtained on the set of extra-urban customers analyzed, it is also observed that the effect of increase in aggregation level is prominent during the time slots when the load pattern follows a trend of increase or decrease in demand, otherwise the probability of change in load is almost the same for all aggregation levels. It can be concluded that the sampling time should be reduced with the increase in aggregation level to preserve information about the behaviour of individual customers. Sampling time of 15 minutes (or even less) provides reasonably good information about the dynamics of individual customers. This approach can lead towards the selection of the metering structure by making the most economical compromise between aggregation level and sampling time. The method presented in this paper provides an overall picture of the time evolution of change in demand. The probabalistic trends obtained may be useful to select possible time slots for initialization of DR programmes and manage demand and supply side resources for economic operation of smart grids and microgrids. The probabilistic analysis has been carried out in this paper on a particular set of aggregated customers. The findings of this research work may differ slightly with the topological and demographic situation. However, the methodology remains generally applicable to other cases. ACKNOWLEDGMENT The research leading to these results has received funding from the European Union Seventh Framework Programme FP7/2007-2013 under grant agreement no. 309048, project
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Wroclaw, Poland – August 18-22, 2014
