# Power Profiling of Smart Grid Users Using Dynamic Time Warping +
## 5 Power Profiling Model
### Load Data Clustering  
Power consumption patterns depend on the consumer's usage behavior, power consumption characteristics of the employed devices, and contextual factors. Consumers use power differently depending on the time of day and day of the week (weekday versus weekend/holiday). For instance, the load factor of a refrigerator is influenced by its consumption patterns, which depend on factors such as thermal load, number of door openings, and opening duration [70]. Compared to a workday, the refrigerator shows different consumption patterns on a weekend/holiday when the number of people at home is different and their behaviors vary (e.g., different sleeping times and more TV watching). Figure 3 depicts the refrigerator's hourly electricity consumption on a workday, a weekend, and over a week, measured every 10 min. Based on this, while the refrigerator's power consumption changes smoothly on a workday, it behaves differently during weekend hours. Typically, a refrigerator exhibits a cyclic pattern of power consumption due to the duty cycling of its compressor and operates at certain intervals during the day [12]. Figure 4 shows the pattern of power usage data for the refrigerator extracted from the AMPds2 dataset. Since the refrigerator operates continuously, its power consumption pattern can be used to profile the user's power usage and identify unusual events. Based on these properties, we select the refrigerator as the source for profiling and evaluate its power consumption to monitor the consumer's power behavior. Figure 3: _Cont._  
Figure 2: ED measurement error to calculate distance between two load data time series with different lengths. As per our power load feature, we consider two sets of power load patterns, namely workday and weekend/holiday load patterns sets. Although these sets of power load patterns can be extended to more categories (e.g., morning or afternoon, day-time or night-time, etc. ), we introduce our profiling model based on these two pattern sets and develop it by adding more categories in the next phase. This binary classification serves as a foundational step for testing DTW-based clustering and evaluating its ability to distinguish meaningful load patterns. Additionally, workday and weekend consumption patterns exhibit statistically significant differences, making them a practical starting point for initial load profiling in smart grids. In this regard, we use the refrigerator's power data points over five months (from September 2013 to January 2014) to extract the consumption features,  
Figure 4: Refrigerator power usage pattern. Figure 3: Refrigerator hourly electricity consumption, measured per 10 min on a typical workday (a), a typical weekend (b), and a typical week (c). train the model, and classify the consumer's power load. We employ the DTW algorithm to measure the similarity between the power usage time series of those days (a total of 22,032 data samples) and set a DTW distance matrix. This matrix shows pairwise DTW distances between power usage data points on different days. To indicate the consumer's usage behavior, we redefine the load factor for a shorter duration, an hour, as the hourly load factor. Based on the daily load factor introduced in Section 4, an hourly load factor (\(lf_{h}\)) is defined as follows:  
\[lf_{h}=(\frac{1}{24}\sum_{k=1}^{24}\frac{1/n\sum_{i=1}^{6}P_{i}}{max(P_{i},1 \leqslant i\leqslant 6)})\times 100, \tag{12}\]  
where \(P_{i}\) is the electrical usage in \(W\) measured of each 10 min period, in total, six power data points in an hour. The hourly load factor is the average of the ratio of hourly power mean to hourly maximum demand per day (among 24 h). To profile power usage, we define criteria based on the DTW matrix of daily usage time series and hourly load factors. Two load profiles are defined: the workday load profile (WoLP) and the weekend/holiday load profile (WeLP).