Abstract
This paper proposes an online framework to characterize demand response (DR) over time. The proposed framework facilitates obtaining and updating the daily consumption patterns of customers. The essential concept of response profile class (RPC) is introduced for characterization and complemented by the measure of the variability in customer behavior. This paper uses a modified version of the incremental clustering by fast search and find of density peaks (CFSFDP) algorithm for daily profiles, considering the multivariate normal kernel density estimator and incremental forms of the Davies-Bouldin (iDB) and Xie-Beni (iXB) validity indices. Case studies conducted using real-world and simulated daily profiles of residential and commercial Chilean end-users have demonstrated how the proposed framework can continuously characterize DR. The proposed framework is proven to achieve realistic customer models for effective energy management by estimating the customer response to price signals at the distribution system operator (DSO) level.
THROUGH deep coordination among grid operators and active customers, the capability of facilities for demand response (DR) and distributed energy resource management will be valuable for ancillary services [
The intrinsic socio-demographic characteristics (individual preferences) of customers and real-time externalities (environmental factors) can influence their response [
Although, in general, there are many studies in the literature about online (or stream) clustering foundations and algorithms (e.g., recent surveys [
However, several studies have recently analyzed the electricity consumption of customers by exploiting important offline clustering methods. For example, [
Despite the valuable contributions of these studies, they lack developing an online characterization. Meanwhile, [
This paper proposes an online framework to characterize the DR from the DSO perspective. The continuous processing of daily profiles makes it possible to know the response profile classes (RPCs) of customers and compute their variability. The framework comprises a modified version of the incremental clustering by fast search and find of density peaks (CFSFDP) algorithm [
The customer response to the price signal has also been studied recently in the literature in the context of DR pricing. For example, in [
The main contributions of the paper are summarized below.
1) An innovative online framework is proposed to characterize the DR. The paper presents the modified incremental CFSFDP algorithm, defined to work in a Hilbert space. Online clustering introduces the multivariate normal kernel density estimator for robustness to the number of objects in clusters and the monitoring of algorithmic performance through the iDB and iXB validity indices.
2) Application of the proposed framework allows the DSO to perform two essential activities: updating the RPC and variability when customer response materializes (at the end of the previous day) and estimating the customer behavior to price signals based on a known RPC (within the current day).
3) The proposed framework is tested with real-world and simulated daily profiles. Results show the online process for obtaining RPCs of residential and commercial Chilean end-users. This paper also provides a comparison analysis with the online algorithm in [
The organization of this paper is as follows. Section II describes the proposed framework and provides theoretical foundations and mathematical models. Section III describes the solution methodology. Section IV presents two case studies with real-world and simulated daily profiles to verify the proposed framework. The performance monitoring and the comparison and sensitivity analysis are discussed in Section V. Finally, Section VI concludes this paper.
The DSO needs to ensure the reliability of distribution system, which may include small distributed solar generation units and generally has tight capacity constraints. By appropriately choosing dynamic price signals to be broadcasted to consumers enrolled in a price-based DR program, the DSO can reduce the distribution system costs and increase its reliability, for example, by shifting flexible consumption to the periods with high stochastic production [
This paper considers the price-setting DSO that aims at managing the demand flexibilities and pursues the estimation of customers’ behavior to price signals. To this end, the DSO performs the daily processing of load profiles through the modified incremental CFSFDP algorithm to update the RPCs and variability of customers. Using the corresponding RPCs, the DSO can generate the expected consumption profiles in response to control price signals. Therefore, it can decide with high certainty the amount of electricity to trade, for example, in the balancing market within the day. The proposed framework, focused on residential and commercial customers, comprises both the previous day, at the end of which the DSO knows the power responses of customers and updates their RPCs and individual variability, and the current day, during which the DSO estimates the customers’ response to a price signal based on known RPCs.
Let designate an expected complex power value to be consumed at time point by a customer under a contract. , where is positioned in the complex plane. It is possible to obtain a bounded and convex approximation of this region given practical bounds, both for active and reactive power. Although the estimation of is fundamental for the analysis at the distribution system level, this paper focuses specifically on the active power responses of end-users. The following linear model for setting a flexible active power profile is defined [
(1) |
(2) |
(3) |
Formula provides the expected response between a minimum and maximum bound for customer at time point . Also, can increase or decrease depending on the market price due to the combined use of shifting and shedding loads. Formula forces ramp limits on the decrease and increase of active power in two successive time points. Finally, a minimum daily energy is specified by (3) to account for basic activities.
An important observation to consider actual consumption features of consumers is that each region is time-varying since the bounds vary over time based on their preferences and environmental factors. This is exploited in this paper by developing an online processing and subsequent characterization of daily load profiles. From the corresponding outcome, it is attainable to differentiate the behaviors of consumers through RPCs, where each RPC represents a portion (of similar daily profiles or vectors) of the polytope that entirely contains the customer’s load scenarios in the vector space. Therefore, based on these classes or portions, a more refined estimation of the consumption activity is feasible.
From a set of daily profiles associated with a RPC, each pair of parameters and of the model can be obtained as the corresponding extreme values, providing the convex (inner) approximation of the active power.
Concerning the values of the maximum ramp rates in , as they are related to the speed at which the consumer can decrease or increase its demand, they differ for each time point of the day and between RPCs. The strategy for its online determination is to consider the load changes from time point to () within the set of daily profiles of the RPC. Then, the following expressions are obtained:
(4) |
(5) |
Formulas and indicate a decrease and an increase in demand concerning the previous time point, respectively.
Lastly, the minimum daily energy can be the lowest total consumption among all the profiles within the RPC.
Daily processing of load profiles favors the appropriate characterization of customer behavior. As consumers can be associated based on the similarity of their consumption patterns, to make this analysis scalable, this paper presents an incremental clustering, which is described in this section. Then, the section considers the performance monitoring and the estimation of the variability of customer response.
Algorithm 1 : online DR characterization |
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Input: set , , , 1: for do |
2: Assign each object to the corresponding cluster by and solve to update the representative |
3: for each cluster , do |
4: Compute the local density by (7) and the minimum distance by (8) and (9) for each object, and select new cluster centers through their products if they exist |
5: Assign each remaining object to the corresponding cluster and select the representatives by (10)-(13) |
6: end for |
7: if new clusters arise then |
8: Compute the minimum distance between all pairs of clusters by (16) and merge accordingly |
9: In the merged clusters, compute the local density by (7) and the minimum distance by (8) and (9) for each object, and select as center the object with the highest product value and the representatives by (10)-(13) |
10: end if |
11: Compute the iDB and iXB validity indices |
12: Update the RPCs and variability of customers by |
13: end for |
Let denote an initial set of power profiles collected during days from customers equipped with SMs, with each vector . Let the initial power profile of each customer recast into daily load profiles. By gathering all these new profiles, the initial set is reformulated then with a total of vectors of -tuples, i.e., , with each vector . Furthermore, a set of load profiles is processed daily after this historical collection.
Any structure in the vector space depends on the similarity metric and the clustering criterion, which expresses how to use the metric. In the paper, the metric , defined in in terms of the norm for any pair of vectors and , is employed. Hence, is a normed linear vector space, particularly a Hilbert space, due to the induced norm [
(6) |
The implementation of the modified incremental CFSFDP algorithm involves the following stages.
The CFSFDP algorithm uses the initial set , where the clustering criterion relies on the computation of two magnitudes for each object , i.e., the local density and the minimum distance concerning the vectors of higher density:
(7) |
(8) |
Unlike [
For the object with the highest density, the distance is:
(9) |
Each cluster center () describes a dominant consumption pattern and is expected to be surrounded by a neighborhood with lower local density and at a relatively large distance from any object with a higher local density. To identify the centers, both the plot of as a function of for each object (the decision graph) and the plot of quantities in decreasing order (each ) can be used [
Each cluster can be represented by a fixed number of representative points generated by selecting well-scattered points and then shrinking them toward the center by a specified fraction. This approach helps to identify the clusters with non-spherical shapes and wide variances in size [
Let be the number of representative points of clusters and be the set of these points to select for any cluster . Their determination follows a sequential order. A selected vector becomes a representative vector . The first representative is:
(10) |
And the rest of them are selected one by one as:
(11) |
To select the second representative, is ; to select the third representative, could be or , and so on. The structure distance is expressed in based on the mean and standard deviation of time point [
(12) |
The shrinking process of the representative points depends on the (user-defined) shrink factor :
(13) |
Shrinking the scattered points toward the center undoes surface abnormalities and mitigates the effect of outliers since these are typically further away from the cluster center [
With a new daily set , each object is assigned to the cluster with the nearest representative. To this end, the local density is set initially to be zero, and the minimum distance is given as:
(14) |
The assignment causes a change in the cluster structure, which requires updating the representative points. Assuming as the nearest representative to the assigned vector , the solution of the problem below is found:
(15) |
where the auxiliary matrix is with the column .
From this result, if produces the maximum value, replaces as the new representative [
In this stage, the algorithm looks for the cluster with more than one dominant pattern. Specifically, in each new cluster , parameters and , and the quantity , are first obtained for each object . Based on these product values, new cluster centers can be identified, and the remaining objects can be assigned as in the CFSFDP algorithm [
This stage happens if new clusters arise, looking for those containing a similar dominant pattern. Specifically, the connected graph is constructed to find the components connected between clusters. The minimum distance between any two clusters and () is computed as:
(16) |
If and , an edge is added between them. After adding all the edges, the graph with multiple components results, and the clusters with the same component can be merged [
Lately, [
For the computation of validity indices, the compactness term is essential. This paper considers the incremental formulation proposed in [
This section presents two case studies to demonstrate the benefits of the proposed framework. The first case study involves residential and commercial electricity data recorded over six weeks (February 1 to March 13, 2020), with 15 min intervals. The Hilbert space is then . The number of customers is 925, charged with a regulated tariff; however, this paper assumes that they practice an optimizing behavior, using electricity in known off-peak periods. Therefore,
Incompleteness in electricity data is a common trend. Then, cleaning is executed by identifying and discarding daily profiles with missing and inconsistent values. The initial set considers the first week of measurement data. Thus, each set of profiles corresponding to the rest of the days is processed incrementally. Likewise, the normalization of each daily profile concerning its maximum value is implemented to facilitate the clustering process.
The application of the CFSFDP algorithm to allows the identification of four initial clusters through the decision graph and the plot of quantities in decreasing order. The results are depicted in
Fig. 1 Plots for identification of initial clusters. (a) Decision graph. (b) Quantities in decreasing order.
Representative points in clusters are the basis for assigning new profiles. The incremental clustering uses the following parameter values: and . To better observe the shrinking of representatives,
Fig. 2 Scatter data of cluster 3 for PCA. (a) Before shrinking process. (b) After shrinking process.
The online algorithm produces five final clusters considering the real-world data set.
Fig. 3 Evolution of clusters for measurement period.
Since customers generally have well-defined behaviors (at least for a specific period) and these behaviors repeat between them, continuous splitting of clusters is uncommon. Also, a splitting rarely generates more than two clusters, which also happens with merging.
Fig. 4 Daily consumption profiles of final clusters in case study 1. (a) Cluster 1. (b) Cluster 2. (c) Cluster 3. (d) Cluster 4. (e) Cluster 5.
Final cluster | Cardinality | Average consumption () |
---|---|---|
1 | 4153 | |
2 | 26684 | |
3 | 2643 | |
4 | 3629 | |
5 | 349 |
The most increased percentage of cluster 2 (over 70%) concludes that this pattern is present in most residential end-users; however, it also includes commercial establishments and small businesses. Cluster 1 presents the highest consumption and more stable behavior. Cluster 3 shows a slightly lower consumption in the morning. Daily profiles with irregular behavior are much more noticeable in cluster 4, and cluster 5 has a typical residential pattern with low consumption.
The appropriate characterization of customer behavior is essential in a control-by-price strategy.
RPC | Number of customers | Combination of RPCs (number of customers of combination) |
---|---|---|
1 | 130 | 2 (122), 4 (4), 1 (4) |
2 | 215 | 1-2 (85), 2-4 (100), 1-4 (2), 2-3 (8), 1-3 (6), 2-5 (12), 3-4 (2) |
3 | 264 | 1-2-3 (58), 1-2-4 (69), 2-4-5 (32), 2-3-4 (91), 1-3-4 (6), 2-3-5 (4), 1-2-5 (4) |
4 | 294 | 1-2-4-5 (14), 1-2-3-4 (248), 2-3-4-5 (29), 1-2-3-5 (3) |
5 | 22 | 1-2-3-4-5 (22) |
To complement this, the variability between the patterns in the combination is fundamental. For example, customers 99 and 630 (which are internal identifications for privacy-preserving) equally have the most common combination: 1-2-3-4; however, the probabilities on energy usage are very different, which are 0.047-0.905-0.024-0.024 and 0.286-0.238-0.286-0.19, respectively, and their corresponding entropy values are and (with the general mean of ), respectively.
The lower value is because of the predominance of the second pattern and the very low probability of the rest, which implies less uncertainty about the daily RPC for customer 99.
Fig. 5 Daily consumption profiles for measurement period. (a) Customer 99. (b) Customer 630.
This case explores the effect of relaxing the regulatory condition to allow the price to fall in off-peak periods and to increase in peak periods.
For simplicity, the DSO broadcasts a single price signal to all consumers each day of the specified week (from March 14 to March 20, 2020); however, customized price signals can be designed according to customer behavior and broadcasted, , each hour of the day to exploit newly available information of system states.
Fig. 6 Price signals generated according to demand of Chilean power system.
The execution of numerical simulations considers the following ideas: ① a realistic situation is applied where consumers use the daily RPC with the highest probability for that day; ② to estimate active power responses to the price signal, customers practice a daily cost minimization; hence, this paper employs the linear programming problem in [
(18) |
For example, considering the same two customers on March 14, the RPC with the highest probability for this day results in RPC 2 for customer 99 and RPC 3 for customer 630.
Fig. 7 Ramp values and , and the minimum and maximum bounds. (a) Customer 99. (b) Customer 630.
After the simulation period, the final clusters increase to six.
Fig. 8 Daily consumption profiles of final clusters in case study 2. (a) Cluster 1. (b) Cluster 2. (c) Cluster 3. (d) Cluster 4. (e) Cluster 5. (f) Cluster 6.
Final cluster | Cardinality (ratio of new daily profiles (p.u.)) | Average consumption () |
---|---|---|
1 | 4227 (0.01) | |
2 | 12003 (0.05) | |
3 | 2467 (0.03) | |
4 | 6041 (0.32) | |
5 | 18960 (0.58) | |
6 | 190 (0.01) | .0 |
The main difference concerning the case with the real-world data set is the new arising cluster 5, related in shape to the previous cluster 2. Most of the new daily profiles under the effect of the price signals are added to this new cluster, which is also indicated in
Lastly,
RPC | Number of customers | Combination of RPCs (number of customers of combination) |
---|---|---|
1 | 9 | 2 (5), 5 (3), 4 (1) |
2 | 67 | 4-5 (34), 2-5 (27), 3-4 (2), 1-2 (1), 1-4 (1), 1-5 (1), 3-5 (1) |
3 | 242 | 1-4-5 (6), 2-4-5 (172), 3-4-5 (29), 1-2-4 (7), 1-3-4 (11), 2-5-6 (7), 1-2-5 (10) |
4 | 305 | 1-2-4-5 (135), 1-3-4-5 (40), 2-3-4-5 (85), 2-4-5-6 (24), 1-2-3-5 (5), 1-2-3-4 (12), 1-4-5-6 (3), 1-2-5-6 (1) |
5 | 282 | 1-2-4-5-6 (17), 1-2-3-4-5 (252), 1-3-4-5-6 (3), 2-3-4-5-6 (10) |
6 | 20 | 1-2-3-4-5-6 (20) |
Considering the real-world data set, this section first analyzes the monitoring of the clustering algorithm based on the iDB and iXB validity indices. The comparison with the online algorithm in [
To assess the cohesion and separation of clusters produced in the online processing,
Fig. 9 Evolution of iDB and iXB validity indices.
On the days when splitting or merging events happen (as shown in
This paper uses the algorithm in [
The algorithm is applied using the norm and the first week for the consensus clustering [
Case | Parameter value | Final cluster | ||||
---|---|---|---|---|---|---|
1 | 7 | 6 | ||||
2 | 7 | 8 |
Fig. 10 Evolution of iDB and iXB validity indices in two case studies. (a) Case study 1. (b) Case study 2.
The sensitivity analysis evaluates the influence of changes on the number of representatives and the shrinking factor . The solutions for a group of selected values of these parameters are given, as shown in
Final cluster | Mean, standard deviation, and difference between the maximum and minimum iDB validity indices | Mean, standard deviation, and difference between the maximum and minimum iXB validity indices | ||
---|---|---|---|---|
5 | 1.57, 0.65, 2.89 | 1.02, 0.65, 2.37 | ||
6 | 2.92, 5.81, 29.68 | 1.52, 0.83, 3.27 | ||
5 | 1.47, 0.30, 1.04 | 1.20, 0.67, 2.23 | ||
3 | 1.27, 0.20, 0.88 | 0.61, 0.31, 1.36 | ||
3 | 1.19, 0.16, 0.76 | 0.60, 0.35, 1.34 | ||
3 | 1.41, 0.52, 3.23 | 1.14, 0.78, 3.50 | ||
5 | 1.28, 0.21, 0.96 | 0.75, 0.42, 1.52 | ||
.0 | 9 | 1.86, 0.19, 0.76 | 1.34, 0.25, 1.22 |
Considering , the trend is to obtain fewer clusters as this number increases. With one representative (the cluster center itself), the result of nine final clusters is obtained, which is close to the eight final clusters of case study 2 using the algorithm in [
Considering , it is not practical to take very low or very high values [
The main idea of this paper is to propose an online framework for DR characterization. In particular, the DSO can use the proposed framework to obtain and update daily the RPCs and variability of customers, and estimate the customer response to a price signal based on a known RPC, which is suitable for effective energy management on the demand side.
Furthermore, the underlying probability distribution of random deviations in demand from expected values is generally unknown. However, the proposed framework contributes to modeling it since each daily deviation can be considered a realization of the corresponding random variable. The use of parameters of this empirical distribution overcomes the limitation of making distributional assumptions that can result in risky or more conservative costly solutions.
Some technical and practical complications arise with each set of load profiles and the higher amount that needs to be processed daily. In particular: ① more computational effort is demanded; ② as clusters extend in the vector space, a larger might eventually be more appropriate for capturing their geometry; and ③ the time to find the information about the RPCs of customers and estimate their responses could be longer than that available by the DSO. Thus, a suitable strategy to be included in the online algorithm is the well-known sliding window [
While processing and analyzing load data streams represent a significant challenge, case studies using real-world and simulated daily profiles of Chilean end-users have demonstrated the applicability of the proposed framework. According to the results, most consumers use four RPCs within the measurement period. Furthermore, the behavior of iDB and iXB validity indices and the comparison analysis have verified the adequate assignment of objects of the online clustering. Future work needs to address the following issues in more depth: ① the impact of dimensionality; ② the definition of the most reliable values for parameters and ; and ③ the correct RPC for estimating the expected response of customers (not necessarily the RPC with the highest probability).
Nomenclature
Symbol | —— | Definition |
---|---|---|
—— | Shrink factor | |
—— | Product of and | |
—— | Interval between two consecutive time points | |
—— | The minimum distance | |
—— | Electricity price at time point | |
—— | Local density | |
—— | Auxiliary matrix | |
, | —— | Initial and non-initial cluster centers of cluster () |
—— | The minimum daily energy of customer | |
—— | Expectation operator | |
, | —— | Clusters of daily profiles, and |
—— | Variability of customer | |
—— | Smoothing parameter of time point | |
—— | Set of customers | |
—— | Total number of vectors of T-tuples | |
—— | Number of daily profiles | |
—— | Number of representative points | |
—— | Probability of cluster center followed up to current day | |
—— | Initial set of daily profiles | |
—— | Set of daily profiles | |
—— | The th initial active power profile | |
—— | Expected active power profile of customer | |
—— | Active power profile of customer | |
, | —— | The minimum and maximum bounds for |
, | —— | Expected active and reactive power profiles of customer l at time point t |
, | —— | Sets of representative points of cluster , and |
, | —— | The maximum ramp-down and ramp-up rates for |
, | —— | Representative vector in and |
, | —— | Structure distances of and |
—— | Consumption region that contains active and reactive power values of customers at time point | |
—— | Set of time points |
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