Applying the Geometric Features of Cumulative Sums to the Development of Event Detection

Abstract

As a result of the severe energy shortage and the greenhouse effect, experts worldwide have been devoted to solving energy management problems. Smart grid construction is an essential technology for mastering energy allocation. Smart grids enable end users to adjust their energy consumption via incentive measures, reduce the frequency of power supply instability, and improve energy efficiency. Non-intrusive load monitoring (NILM) is a vital technology for smart grid construction. One of the fundamental steps of NILM is event detection. Proper event detection can increase the accuracy of load identification. Among traditional methods, especially the event detection method developed with the CUSUM method, although the accuracy is reasonable, the precision, recall, and f₁ score are not relatively better. Thus, there is an opportunity to improve the performance of CUSUM. Additionally, many studies focus on the step-like event, but the long-transient event is often overlooked in event detection. Therefore, in this study, it was observed that when the transient current deviates from the steady-state current, the transient current can be regarded as a key indicator for event detection. With this observation, a method is proposed to convert the root mean square (RMS) current into a cumulative sum (CUSUM) diagram method and identify turning points representing events from the CUSUM geometry. Once the slope of the turning point has been determined, event detection is achieved. Compared with traditional methods, the proposed method is easy to implement, its recognition rate can reach around 98%, and the window length is reduced from 5 s to 3 s.

1. Introduction

China’s power rationing policy in 2021 influenced the operation of numerous domestic enterprises, and multiple countries worldwide have faced an energy crisis because of the increasing price of natural gas. Power supply stability is a critical problem in maintaining the balance between the supply and demand for power. To maintain power supply stability, numerous experts have actively promoted the demand response policy, which is currently implemented by numerous countries worldwide to achieve power supply stability. Proposed in the 1980s, non-intrusive load monitoring (NILM) is a vital technology used in demand response [1]. The purpose of NILM is to identify individual loads by analyzing aggregate power characteristics. NILM enables the collection of power consumption data for electrical appliances operated by the end user. In the case of an insufficient spinning reserve, NILM technology can be used in conjunction with demand response measures to achieve a situation mutually beneficial to power supply and demand balance. Figure 1 depicts the typical architecture of an NILM system. Event detection is used to conduct a complete NILM procedure. Because numerous experts have been devoted to NILM research, many load characteristics, such as $I_{rms}$ [2], the V-I trajectory [3], grayscale images [4,5], and the first-order derivative [6] have been developed, and novel event detection techniques have been proposed.

In this study, an event detector was developed using the geometric features of RMS current variations in cumulative sums (CUSUMs). The following are the main contributions of this study.

• VERSATILITY: Some research results are only verified by using a single dataset [7,8]. Because the results are not objective, it is difficult to prove that the proposed method could be applied to other datasets. Therefore, the proposed method was validated using two public datasets, namely the Controlled On/Off Loads Library (COOLL) [9] and Plug Load Appliance Identification Dataset (PLAID) [10], and a private dataset. In terms of versatility, the proposed method can be easily applied to other datasets.
• SIMPLICITY: The traditional CUSUM methods need to repeat the detection of data several times, which requires nest structures [11] or the assistance of hardware equipment [12] to confirm the correctness of the identification results. The proposed method does not require complex mathematical calculations because the fundamental slope and numerical comparison concepts can be easily implemented.
• THE ENHANCEMENT OF ACCURACY AND EFFICIENCY: The accuracy needs to be improved in the literature [11,12,13] that used CUSUM for event detection. In addition, the window length of each event was also shortened from 5 s to 3 s, and the impact of sampling frequency on event detection was also investigated. Therefore, to perform effective identification and reduce misjudgments under various conditions such as steady state, transient, or noise during the event detection process, the CUSUM geometric features as the basis were used by the proposed method for event detection, thus improving the accuracy and efficiency of identification.

This paper is organized as follows. Section 2 describes related studies. Section 3 presents the data preprocessing and geometry of the CUSUM plots. Section 4 describes the proposed event detection algorithm. The evaluation metrics used in this article are shown in Section 5. Section 6 compares the results of various methods via validation against the COOLL dataset, the PLAID dataset, and a private dataset. Lastly, Section 7 concludes the study.

2. Related Studies

According to the works of the literature, numerous novel methods for NILM event detection have been proposed. These methods can be classified into machine learning (ML), numerical, and statistical probability methods.

2.1. Event Detection Methods

As computational speeds have increased, ML has become more prevalent in various research fields. Density-based spatial clustering of applications with noise (DBSCAN) is an ML algorithm used for event detection [7]. In this study, time-series data, including steady-state and transient-state data, were classified to determine whether an event occurred. However, to use DBSCAN, the parameters Eps (area density) and MinPts (number of points in the neighborhood) must be set differently for each dataset. In [4,14,15], convolutional neural network pattern recognition was used for event detection. However, neural networks require numerous images for training. In [8,16], a multi-dimensional BIRCH clustering (DNB) technique and shapelet-based event detection were proposed to classify NILM events. But the evaluation metrics show that the results can still be improved. Other ML algorithms—such as k-means clustering [15] and the principal component analysis of active, reactive, and distortion approaches proposed in [17]—have also been used for NILM event detection. Generally, the benefits of ML for event detection could be achieved possibly without the need to preset thresholds.

In [18], a method called frequency-invariant transformation of periodic signals was proposed for optimizing signals for event detection. Its purpose is to separate the aggregated current and regulate it into its individual periods by using voltage as a reference, but the signal after conversion may still be distorted in some cases [8]. In other methods, active power changes are examined when the system is activated. As reported in [19,20,21], ∆P in Equation (1) can be used to determine the amount of change in active power:

$$\mathsf{\Delta }P=P_i-P_{i-1}$$

(1)

where $P_i$ indicates the active power value at time t and $P_{i-1}$ indicates active power at time t − 1. ∆P is thus a method used to calculate the difference in active power for a given window before versus after an event. The selection of the window duration and threshold affects the accuracy of event detection. In [5], an event judgment method based on derivative analysis was proposed. This method describes changes in the RMS current in terms of derivatives, which represent the occurrence of an event. However, this method could treat noise as an event for noise derivatives. In [22], the turn-on events are detected by applying an adaptive threshold. One of the advantages is that the spike could be filtered out by the proposed method, but the result is not better than [6]. In [23], the threshold boundary is between 10 W and 60 W. This method still needs to be improved in product diversity. Specifically, all methods mentioned in this paragraph can be categorized as numerical methods.

CUSUM is a statistical probability method used for NILM event detection. In [11], an improved CUSUM method added bootstrapping algorithm was proposed for detecting likely events using confidence levels rather than by setting thresholds. However, in this approach, each action requires numerous resampling instances. In [12], a method combining Kalman filtering with an improved CUSUM algorithm was proposed. To increase the accuracy of event detection, especially during AC start-up, the authors optimized the values of the upper and lower limits by using adaptive factors. This process means that the judgment criterion of determining event occurrence is relatively flexible without being affected by fixed values, thereby increasing the accuracy.

In summary, the methods discussed above still have their respective disadvantages. ML methods need substantial computational time and hardware resources, as well as require numerous pictures to achieve neural network training. Regarding numerical methods, the literature [19,20,21] that only uses power changes shown in Equation (1) as the basis for whether an event occurs is insufficient because the power value will vary when noise appears. For the probability method, the research results of past literature show that the accuracy of event detection using CUSUM was around 91%. Although CUSUM is a traditional method, it still has room for improvement. Additionally, most methods can detect step events but perform poorly in long transient events [12,18,22,24,25]. Therefore, an event detection method was proposed, applying geometric features of cumulative sums. The judgment conditions of the proposed event detection method not only consider the states generated during the signal transmission process (such as steady state, transient state, long transient state, noise, etc.) to improve the detection accuracy but also have the advantageous characteristics of not requiring a large number of hardware resources to complete event detection.

2.2. Event Detection Methods with CUSUMs

Several CUSUM-based event detection approaches have been proposed [11,12,13]. As shown in

Table 1. Accuracy comparison of existing event detection methods with improved CUSUMs.

Paper	Improved Method	Dataset	Accuracy
[11]	Based on CUSUM and adding bootstrapping	Private dataset	Above 91% at conclusion
[12]	Adaptive factors @ CUSUM	Private dataset	N.A.
[13]	CWA @ CUSUM	Industrial load dataset	Above 77% mentioned in the table 2 of [13]

, for event detection in any database, increasing the degree of accuracy is difficult if only a pure CUSUM method is used. In other words, CUSUMs must be improved to enhance their detection capability.

For example, the authors in [12] mentioned that “In the traditional CUSUM algorithm, the limit value is fixed during iteration...and fixed parameters in Formula (2) will fail to detect the whole long transient time in the special event such as the AC start-up”.

$$\left\{\begin{array}{c}{ g}_i^{+}=\left(0,g_{i-1}^{+}+\left(s_i-s_{i-1}\right)-\beta \right), { g}_0^{+}=0\\ g_i^{-}=\left(0,g_{i-1}^{-}-\left(s_i-s_{i-1}\right)-\beta \right), g_0^{-}=0 \end{array}\right.$$

(2)

where $g_i^{+}$ and $g_i^{-}$ are the positive and negative CUSUM values and $\beta$ is the lower event limit”.

In Equation (2), $\beta$ is a fixed value that restricts both $g_i^{+}$ and $g_i^{-}$. Therefore, the key to enhancing accuracy is to use an adaptive judgment value rather than a fixed value. A similar concept was discussed in [13,26].

Many mathematical approaches have been proposed for increasing the accuracy of NILM event detection. In this study, a mathematical approach based on the geometric features of CUSUMs was proposed. In the following section, the proposed approach is described and analyzed in detail.

3. Data Preprocessing and Geometry of CUSUM Charts

In NILM, event detection may influence the correctness of load identification. In this section, the proposed event detection algorithm is discussed in detail.

3.1. AC Transient Current Behavior of Different Appliances

Most appliances are equipped with electronic components whose purpose is to save energy. When an appliance is activated or deactivated, it exhibits a unique instantaneous transient current. Figure 2 shows the AC waveforms of four appliances. The transient current in each appliance is circled by a dotted red line. Although each appliance has distinct transient current characteristics, these characteristics are collectively referred to as transition behavior. This phenomenon also represents the occurrence of an event.

3.2. The Behavior of RMS Current in Different Appliances

If the power of the direct voltage, which is applied to the same resistance as the alternative voltage, is the same as the average power of the alternative voltage, the direct current equivalent voltage is the RMS value of the alternative voltage. The following is the mathematical formula for RMS voltage value if N alternative voltage signals are sampled in one cycle:

$$v_{rms}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{v}_i^2}$$

(3)

where $v_i$ represents $i_{\mathrm{t}\mathrm{h}}$ alternative voltage value.

To visualize the behavior of the current, the alternative current (AC) value (Figure 2) can be converted into RMS current (Figure 3). As shown by the RMS current curve, the value of the current jumps from the first steady state to the second steady state. This process was also described by using a first-order derivative in [6] to determine whether an event occurred.

3.3. CUSUM Plot of RMS Current

CUSUM is a time-series data analysis method proposed by E. S. Page of Cambridge University in 1954. This method has been widely used to detect abnormal data points in relatively stable data series. In this study, CUSUM geometry was used as the basis for event detection. As shown in Figure 4, which depicts CUSUM charts converted from Figure 3, when the time series data are abnormal, a turning point is observed. Therefore, the RMS current value can be converted into a CUSUM graph for event detection.

3.4. Characteristics of CUSUM Plots under Various Conditions

(1)

Step-like Event

Figure 5b,d depicts CUSUM plots converted from RMS current plots (a) and (c). As shown in Figure 5a,b, when the appliance is energized, the turning point is at the 18th point on the RMS current and CUSUM charts. As shown in Figure 5c,d, when the appliance is de-energized, the turning point is at the 13th point on the RMS current and CUSUM charts. Because this turning point is the CUSUM extreme point, the turning point depicted on the CUSUM chart represents event occurrence.

(2)

Long Transient Event

For some appliances, the RMS current values do not all exhibit a step-like state after activation. Figure 6a depicts a post-activation inrush current, marked by a dotted red line; after this, the current returns to a value close to that before the appliance was activated. The current then slowly increases until the next steady state. In Figure 6b, which is a CUSUM chart of Figure 6a, the red circle highlights a turning point that represents a state change. Therefore, regardless of whether the event is a step-like or long transient event, the judgment method used to search for the turning point is the same in both CUSUM plots.

(3)

Transient Current State

Following an event, the operating current of some household appliances tends to reach a steady state by either slowly increasing, as shown in Figure 7a, or slowly decreasing, as shown in Figure 7c. This current behavior closely resembles a quadratic curve on a CUSUM plot, as shown in Figure 7b,d. Regardless of whether the current increases or decreases, an extreme point exists on both CUSUM plots. This common point is not only a CUSUM geometric feature but also a key point for event detection.

(4)

Noise

As shown in Figure 8, because noise randomly appears in the circuit, the influence of noise on event detection must not be neglected. In [17], a Kalman filter and improved CUSUM algorithm were used to filter noise and thus prevent event detection misjudgment. However, in the event detection method proposed in [7], noise is preserved without any special preprocessing. In the following section, the method used to avoid misjudging noise as an event is described in detail.

4. Proposed Event Detection Algorithm

This section first introduces existing mathematical methods and then discusses the role of each part of the proposed event detection technique in detail.

4.1. Major Mathematical Methods

• CUSUM

For successive samples $x_1$, $x_2$, $x_3$,…, $x_n$, if the target value of the average number of processes in the controlled state is ${\mu }_0$, the CUSUM is calculated as follows:

$${\mu }_0=\frac{\sum _{i=1}^n{x}_i}{n}$$

(4)

$$\begin{array}{c}{C}_1=(x_1-{\mu }_0)\\ C_2=(x_2-{\mu }_0)+C_1\\ C_3=(x_3-{\mu }_0)+C_2\end{array}$$

$$C_n={\sum }_{i=1}^n(x_i-{\mu }_0)=(x_n-{\mu }_0)+C_{n-1}$$

(5)

Once the CUSUMs $C_1$, $C_2$, $C_3$, …, $C_n$ are calculated, the CUSUM chart or plot can be completed, where the sample number i is the horizontal axis and CUSUM $C_i$ (i = 1, 2, 3, …, n) is the vertical axis.

• Mathematical formulas for event detection

When a step-like or long transient event occurs, the CUSUM curve is similar to a quadratic curve. Because the extreme point determines the location of the event, detecting this extreme point is a crucial step. The following judgment formula is used to detect the extreme point.

In a CUSUM set in which $C_i\in$ {$C_1$,$C_2$,$C_3$, …,$C_n$}, if an energizing event occurs and $C_i$ is the turning point, $C_i$ must satisfy the following constraint:

$$(C_i<C_{i-m})\cap \left(C_i<C_{i+m}\right), \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{m}\in N.$$

(6)

In contrast, if a de-energizing event occurs and $C_i$ is the turning point, $C_i$ must satisfy the following constraint:

$$(C_i>C_{i+m})\cap \left(C_i>C_{i-m}\right), \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{m}\in N.$$

(7)

These two constraints are derived from the characteristics of CUSUM geometric figures by using the COOLL and PLAID databases; more than 4500 events and non-events were generated from these two databases.

4.2. Event Detection Flow of the Proposed Method

Figure 9 depicts the proposed event detection method, with the CUSUM plot as the event detection object. The following is a detailed description of each step and the corresponding mathematical formula. Meanwhile, the pseudo-code is also listed for the discussion of computational complexity.

(1)

The Explanation of Pseudo-codes and Time Complexity

• Algorithm 1 is the main body of the program architecture. It is responsible for data preprocessing and execution of proposed methods.

Algorithm 1: Main Program

1.       while (true) 2.           Input: voltage_ac, current_ac; 3.           i_rms = create_rms_current(voltage_ac, current_ac);                       // Algorithm 2 4.           noise_pt = noise(i_rms);                                                                      // Algorithm 3 5.           if (noise_pt == 0) 6.               if (i_rms(:) > ${10}^{-3}$) 7.                   Irms_M ← max(i_rms(:)); 8.                   Irms_m ← min(i_rms(:)); 9.                   if (((Irms_M - Irms_m) / Irms_m) > ${\mathrm{i}}_{\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}}$) 10.                       i_rms_cs ← CUSUM(i_rms);                                                  // Algorithm 4 11.                       for cs_x = 2: ${\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}}_{\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}-1}$                                              // search extreme points 12.                           if (i_rms_cs(cs_x) > i_rms_cs(cs_x − 1)) and 13.       (i_rms_cs(cs_x)> i_rms_cs(cs_x+1)) 14.                               max_l ← $\left|\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x})-\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x}-1)\right|$ 15.                               max_r ← $\left|\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x})-\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x}+1)\right|$ 16.                           end 17.                           if (i_rms_cs(cs_x) < i_rms_cs(cs_x − 1)) and 18.       (i_rms_cs(cs_x) < i_rms_cs(cs_x+1)) 19.                               min_l ← $\left|\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x}-1)-\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x})\right|$ 20.                               min_r ← $\left|\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x}+1)-\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}\_\mathrm{c}\mathrm{s}(\mathrm{c}\mathrm{s}\_\mathrm{x})\right|$ 21.                           end 22.                       end 23.       if ((max_l >${\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{\mathrm{t}\mathrm{h}}$) or (max_r > ${\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{\mathrm{t}\mathrm{h}}$) or                                              // judge event or nonevent 24.                               (max_l > ${\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{\mathrm{t}\mathrm{h}}$) or ( max_r > ${\mathrm{S}\mathrm{l}\mathrm{o}\mathrm{p}\mathrm{e}}_{\mathrm{t}\mathrm{h}}$)) 25.                               print(“An event occures”); 26.                       else 27.                               print(“Non-event occures”) 28.                       end 29.               end 30.           end 31.     end 32. end

Steps 2 to 4 and steps 7 to 8 take constant time to read. The total complexity time is 5*O(1). Step 11 to 22 takes O(n) and 2*O(1) from step 23 to 27. So, the overall time complexity of the algorithm is 7*O(1) + O(n).

• In Algorithm 2, the AC signal is converted into the RMS current value.

Algorithm 2: Create RMS Current

1.       Input: voltage(:), current(:) 2.       for i = 1: ${\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}}$ 3.           if (voltage(i) < 0 and voltage(i+1) >= 0) 4.            data_i(i,:) ← current(:) // start to record sampling points for every cycle 5.           end 6.       end 7.       for i = 1: ${\mathrm{d}\mathrm{a}\mathrm{t}\mathrm{a}\_\mathrm{i}}_{\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}}$ 8.         i_rms(i) ← rms(data_i(i,:)) 9.       end 10.       return (i_rms)

Step 1 takes constant time to read voltage and current, 2*O(1). In steps 2 to 6, the for-loop performs the arithmetic operation to calculate the consecutive samples with the complexity of O(n). In steps 7 to 9, another for-loop performs the arithmetic operation to calculate the consecutive samples with the complexity of O(m). The overall time complexity of the algorithm is 2*O(1) + O(m)+ O(n), where m > n.

• In Algorithm 3, it is shown how noise is ruled out.

Algorithm 3: Noise

1.   Input: i_rms 2.   for i_cnt = 1: ${\mathrm{i}\_\mathrm{r}\mathrm{m}\mathrm{s}}_{\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{x}}$ 3.       ${\mathrm{i}}^{\prime }(\mathrm{i}\_\mathrm{c}\mathrm{n}\mathrm{t})$ ← $\left|{\mathrm{i}}_{{\mathrm{i}}_{\mathrm{c}\mathrm{n}\mathrm{t}}+1}-{\mathrm{i}}_{{\mathrm{i}}_{\mathrm{c}\mathrm{n}\mathrm{t}}}\right|$ 4.   end 5. // $1<\mathrm{m}<\left(\mathrm{t}{\mathrm{p}}_{\mathrm{n}\mathrm{o}}-\mathrm{k}\right)<\mathrm{t}{\mathrm{p}}_{\mathrm{n}\mathrm{o}}<\left(\mathrm{t}{\mathrm{p}}_{\mathrm{n}\mathrm{o}}+\mathrm{k}\right)<\mathrm{p}<\mathrm{n}$ 6.   for cnt = m: (tp_no-k) 7.       current1← current1 + ${\mathrm{i}}_{\mathrm{c}\mathrm{n}\mathrm{t}}$ 8.   end 9.   for cnt = (tp_no+k): p 10.       current2← current2 + ${\mathrm{i}}_{\mathrm{c}\mathrm{n}\mathrm{t}}$ 11. end 12. $\mathsf{\Delta }\mathrm{i}$ ← $\left|\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}1-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}2\right|$ 13. if (${\mathrm{i}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\prime }$ > ${\mathrm{i}}_{\mathrm{t}\mathrm{h}}$ and $\mathsf{\Delta }\mathrm{i}$ <${\mathrm{i}}_{\mathrm{t}\mathrm{h}}$) 14.       noise_ptr ← 1 15. else 16.       noise_ptr ← 0 17. end 18. return (noise_ptr)

Step 1 takes constant time, O(1) to read voltage and current. In steps 2 to 4, 6 to 8, and 9 to 11, the for-loop performs the arithmetic operation three times to calculate the consecutive samples with the complexity of 3*O(n). Step 12 takes constant time to read $\mathsf{\nabla }i$, O(1). The same procedure is conducted from steps 13 to 17, O(1). So, the overall time complexity of the algorithm is 3*O(1) + 3*O(n).

• In Algorithm 4, CUSUM is generated by RMS current.

Algorithm 4: CUSUM

1.    Input: rms-current 2.    u_mean = mean(rms-current) 3.    temp = 0 4.    for i = 1: ${\mathrm{r}\mathrm{m}\mathrm{s}-\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{x}\mathrm{m}\mathrm{a}\mathrm{x}}$ 5.      temp ← temp + rms-current(i)-u_mean 6.      cusums(i) ← temp 7.    end 8.    return (cusums)

Steps 1 and 3 take constant time, O(1) to read rms-current and temp. Step 2 employs the average of numbers with time complex O(n). In steps 4 to 7, the for-loop performs the arithmetic operation to calculate the consecutive samples with the complexity of O(n). The Overall time complexity of the algorithm is O(1) + 2*O(n)

The time complexity of the four algorithms are 7*O(1) + O(n) + 2*O(1) + O(n) + O(m) + 3*O(1) + 3*O(n) + O(1) + 2*O(n). The major contributors in this time complexity calculations are the RMS current and the loops where it calculates the difference. Because m is much greater than n and 4, the time complexity of the proposed method is dominated by O(m).

(2)

Noise removal (step A1)

Noise is a momentary signal. As shown in Figure 10a, after a noise signal disappears, the current returns to its original value before the emergence of the noise. Therefore, noise events should not be neglected and need to be removed appropriately.

Figure 10b depicts the first-order derivative generated from Figure 10a. This derivative provides two essential clues for judging a noise event. The first clue is the presence of turning points circled by dotted green lines. The second clue is that the locations of the current value near the turning points denoted by solid red lines are almost the same. The two constraints used to exclude a noise event are sequentially defined as follows.

To define the first constraint, assume that RMS current is defined as $i_i\in$ {$i_1,i_2$,$i_{3,}$...$, i_n$} and that its corresponding first-order derivative is $i_k^{\prime }\in$ {$i_1^{\prime }$, $i_2^{\prime }$,$i_3^{\prime }$,…,$i_{n-1}^{\prime }$}. The first constraint used to define noise is

$$\left|i_k^{\prime }\right|\ge i_{th}^{\prime }\cap \left|i_{k+1}^{\prime }\right|\ge i_{th}^{\prime }, \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} 1\le k\le n-1.$$

(8)

To define the second constraint, assume that the location of the RMS current value near the turning points is almost constant:

$$\mathsf{\Delta }i={\sum }_{i=m}^{i=t{p}_{no}-1}{i}_i-{\sum }_{i=t{p}_{no}+1}^{i=p}{i}_i\le i_{th},$$

(9)

where $1<m<tp\_no<p<n$, and $tp\_no$ is the time point at which the turning point occurs. For example, the values of tp_no are 14 and 15 in the example shown in Figure 10b. The time number of sampling points around the turning point are 8, 9, 10, 11, and 12 for m and 17, 18, 19, 20, and 21 for p. When the two constraints are met (i.e., $\left|i_k^{\prime }\right|\ge i_{th}^{\prime }$, $\left|i_{k+1}^{\prime }\right|\ge i_{th}^{\prime }$, and $\Delta i$ $\le$ $i_{th}$), noise is detected, and processing is stopped until the next step.

Because the accuracy of event detection is affected by noise, instead of eliminating noise, the proposed method prevents noise from being misclassified as an event.

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Confirming the Variety of Operating Current (step A2)

Although the change in current of low-power appliances (e.g., light bulbs) is negligible in the context of demand response, the magnitude of this current must nonetheless be confirmed because this small-magnitude current still represents the energized state of an appliance. In Figure 11a, sliding windows are illustrated by red and green rectangles. The sliding window duration and time interval are assumed to be 15 and 5 s, respectively. As shown in Figure 11b, the current denoted by the red rectangle is the first event, and the distribution is smaller than ${10}^{-3}$ A.

Because this situation is not ruled out by Equation (10), a turning point, shown as a dotted line on the CUSUM chart in Figure 11c, can be observed, and this point represents the occurrence of an event. In this case, a trace current can be observed from 0 to 20 s, and its distribution does not represent event occurrence. To determine whether all acquired current values are suitable for event detection, a threshold must be set. It can be described by Equation (10).

$$I_i \le I_{th}$$

(10)

where $I_i$ is the set of sampled values within the range of a sliding window and 1 $\le$ i $\le$ M. M is the total number of points in a cycle.

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Confirming the Operation Status (step A3)

During operation, the current value does not remain constant. Figure 12a depicts the RMS current waveform of an energized appliance, and it appears to be flat. However, according to the current scale and current values shown in Figure 12b, the current is not the same at each time point. As illustrated in Figure 12c, when the RMS current value is converted into a CUSUM, two turning points in red circles are observed. Because the selected current is already in a steady state, these points are false events.

In Figure 12b, $I_{max}$ and $I_{min}$ are the maximum and minimum values, respectively, of RMS current in the sliding window. As indicated earlier, Equation (11) is used to exclude the following condition:

$$\mathsf{\Delta }I=\frac{I_{max}-I_{min}}{I_{max}}\le I_{ratio}.$$

(11)

Therefore, according to Equation (11), a steady state can be confirmed in advance, and the occurrence of a non-event can also be simultaneously excluded.

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The Operation of CUSUM (step A4)

Before the operation of CUSUM, in order to obtain ${\mu }_0$, the first step is to calculate the average of all data within the range of sliding window by Equation (4). Then, a CUSUM set will be obtained by Equation (5).

(6)

Turning point selection of CUSUM (step A5)

The RMS current that passes the three aforementioned restrictions (A1–A3) does not belong to any of the following three categories: noise, trace current, and fluctuating load current. In this step, because the local extremum denotes the turning point and represents event occurrence, Equations (6) and (7) are used to determine the local extremum. However, the local extremum can be simply obtained via comparison, and the RMS current is then converted into a CUSUM graph. As shown in Figure 13, the two local extreme points satisfy Equations (6) and (7), respectively, indicating the occurrence of two events. In summary, this simple and intuitive method relies on the geometric features of CUSUMs.

(7)

Event Judgment (step A6)

According to Figure 14a,b, not all CUSUM graphs have only one extremum. Although the two turning points in the graph are continuous, one represents an event, and the other does not. In Figure 14b, the first turning point, highlighted by a green circle, indicates an energizing event, which is shown by a green ellipse in Figure 14a. Although another local maximum is observed around the green rectangle in Figure 14b, this process is merely a transient process, shown as a green rectangle in Figure 14a. This behavior is readily observed in Figure 7. Therefore, to determine whether an event occurs, a criterion must be set by using Equation (12) for the flowchart displayed in Figure 9.

The slope of the turning point, denoted as ${Slope}_{TurningPoint}$, must satisfy the following threshold:

$${Slope}_{TurningPoint}\ge {Slope}_{th}.$$

(12)

Figure 15a depicts another case of consecutive events. The event shown by a red circle represents the first event, which is an energizing event. The event indicated by a green circle represents the second event, which is a de-energizing event. The CUSUM plot shows two turning points, which are regarded as two consecutive events because both of them meet the restriction of Equation (12). Although the proposed event detection method can identify multiple events, it focuses on whether any event has already occurred. This topic was similarly discussed in [6], in which a certain time threshold $T_{th}$ was set to eliminate the remaining events, which were treated as false events.

In Equation (12), ${Slope}_{TurningPoint}$ is an essential parameter. As illustrated in Figure 16a,b, because each turning point is a local extremum, each turning point has two points adjacent to it (indicated by the red arrows). Therefore, each turning point and its two adjacent points generate two slopes, with the larger slope being selected. If the value of this slope exceeds the threshold of Equation (12), an event is concluded to have occurred. This process confirms event occurrence.

According to the aforementioned assumption, once the maximum slope is calculated within the window length, the other slope value is eliminated, and no certain time threshold is set.

So far, the proposed flow of event detection and time complexity were fully described. Before the CUSUM plot is generated for analysis, the detected RMS current must satisfy three constraints. Because not all turning points on the CUSUM plot represent events, once the extreme point search process in the CUSUM plot was completed, the slope of the extreme point must meet the criteria presented in Equation (12) to correctly detect a reasonable event.

5. Evaluation Metrics

Accuracy, Precision, Recall, and $F_1 score$ are indicators that are often used to evaluate the accuracy of system performance. The calculation formulas for these four types of indicators are as follows:

$$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$$

(13)

$$Precision=\frac{TP}{TP+FP}$$

(14)

$$Recall=\frac{TP}{TP+FN}$$

(15)

$$F_1=\frac{2\times Recall\times Precision}{Recall+Precision}$$

(16)

where TP is true positive, TN is true negative, FP is false positive, and FN is false negative. Accuracy is the precision rate, which represents the ability of the model to correctly judge the authenticity in all situations. Precision is the accuracy rate, which represents the ability of the model to distinguish real events. Recall is the recall rate, which represents the ability of the model to detect real events. $F_1$ is a comprehensive evaluation index, which represents the comprehensive ability of the model to detect and identify real events. In this study, the performance of the proposed algorithm is also evaluated according to these four types of indicators after event detection.

6. Experimental Results

To verify the performance of the proposed method, a private dataset (2 kHz sampling frequency) and two public datasets (COOLL/100 kHz sampling frequency and PLAID/30 kHz sampling frequency) were used in the experiment. The power of appliances is between 1 W and 2700 W.

Table 2. Parameters of the algorithms.

Paper	Description	Parameter Value
[7]	Min. pts, Eps	5, 0.1
	Window length	15 s.
	Threshold of power different of event	$P_{th}$ = 15 W
	Interval time	$t_{th}$ = 2 s.
[11]	Bootstrapping by confidence level	95~98%
	Min. MSE.	Equation (5) proposed in [11]
	Power range	$P_{th}$ = 60 W~1800 W
Proposed method	Noise Creation	$i_{th}^{\prime }$ > 0.5, $i_{th}$ < 0.1
	Magnitude of the operating current	$I_{th}$ ≥ 0.015 A
	Current variation	$I_{ratio}$ = 3%
	Event judgment	${Slope}_{th}$ = 0.06
	Window length	3 s.
	Interval	1.5 s.
	Power range	$P_{th}$ = 10 W~2700 W

lists the parameters of the proposed method and other existing methods.

Table 3. Metrics comparison by different methods (%).

Paper	Method	Dataset	Accuracy	Precision	Recall	${\mathit{f}}_1$ Score
[7]	DBSCAN	COOLL	81.27	90.12	72.24	83.88
		PLAID	70.25	94.85	57.27	71.42
[11]	Traditional CUSUM	COOLL	67.42	67.90	78.06	80.24
		PLAID	76.66	85.93	76.58	80.99
Proposed method		COOLL	94.52	94.53	97.53	96.00
		PLAID	97.30	98.59	97.23	97.90

presents the test results obtained using the proposed method and other methods with the two public datasets.

Table 4. The comparison with different sampling rates.

Sampling Rate	Cooker (700 W/110 V 60 Hz)	Vacuum (220 W/110 V 60 Hz)
20 K s/s
10 K s/s
5 K s/s
2 K s/s
1 K s/s
500 s/s
200 s/s
100 s/s
50 s/s
20 s/s

shows the experimental results of a simulated case involving common household appliances. All experiments were conducted on MATLAB, and the results of each event detection will be verified based on the actual status, including events and non-events, to confirm the correctness of the event detection results.

6.1. The Selection of Slope Threshold

To determine the occurrence of an event, the slope threshold for the turning points needs to be defined. Therefore, to objectively obtain the suitable threshold, an experiment to obtain four metrics values (accuracy, precision, recall, and f₁ score) via different slopes was conducted on the COOLL dataset. The results are shown in Figure 17.

If the values of these four evaluation indicators differ little from each other, this indicates that the event detection results have high confidence. Therefore, the minimum value of mean square error (MSE) shown in Equation (17) is used to obtain the appropriate value for the slope threshold setting. After calculation, the distribution of MSE values is shown in Figure 18. The minimum MSE value is 0.1742, and the corresponding slope value is 0.06. In other words, 0.06 is the best option for the slope threshold definition to judge whether an event occurs.

$${ \mathrm{Y}=(Accuracy-1)}^2+{(Precision-1)}^2+{(Recall-1)}^2+{ (f_1-1)}^2$$

(17)

6.2. Metrics Comparison for Various Methods

Several CUSUM-based event detection methods have been proposed in [11,12,13,27]. However, in this study, for the sake of brevity, only studies reporting high accuracy were selected for comparison. Table 2 lists the parameters and methods used in each study. For a fair comparison, each method uses the COOLL and PLAID datasets as test sources.

• For the experiment using parameters in [7], the evaluation index (precision) shows that non-events predicted to be events are less (90~94%), but (recall) shows that events predicted to be non-events are more (57~72%). Finally, the f₁ score shows that the experimental results of the two public datasets are very different (71~83%). Therefore, it can be seen that the parameters in [7] could not be applied to different datasets.
• For the experiment implemented by the method in [11], the evaluation index (precision) shows that non-events predicted to be events are more (67~85%), and the index (recall) shows that events predicted to be non-events are more (76~78%) as well. Finally, the f₁ score is around 80%. Thus, the method [11] is not less affected by different datasets.
• For the proposed method, the evaluation index (precision) shows that the number of non-events being predicted as events is low (94~98%), and the index (recall) shows that the number of events being predicted as non-events is also low (around 97%). Finally, the f₁ score shows that the experimental results conducted by the two datasets have a small difference (96~98%). Hence, the parameters of the proposed method are also applicable to different datasets.
• The slope used to determine whether an event occurred is slightly smaller than the threshold, resulting in misjudgment (about 1%).
• The appliances with power less than 10 W are also easily misjudged (around 1~2%). In order to improve the accuracy of judgment, the proposed method is more suitable for event detection of electrical appliances with power above 10 W.

In summary, the metrics comparison for the proposed method (slope threshold = 0.06) and other existing methods are listed in Table 3. It shows that the proposed method had better performance than it did in [7,11].

6.3. The Effect of Different Sampling Frequency on Event Detection

In this section, the comparison of the AC current measured by the two electrical appliances with different sampling rates is shown in Table 4. Each picture was measured for 6 min with power on at the first minute and power off at the fifth minute. The captured current for these two electrical appliances starts to distort severely when the sampling frequency is reduced to 20 s/s, implying that the resolution of the sampling rate must be greater than 20 s/s. However, when the alternative currents are converted to RMS currents, the proposed method is conducted (window length 3 s, interval 1.5 s), and three events, in the order of an event, a non-event, and an event, should be detected (Figure 19). To demonstrate the extent of the distortion, the RMS current values of non-event were analyzed with Formula (11) (($I_{max}-I_{min}$)/$I_{max}$), resulting in the distribution of steady-state current changing with the sampling frequency, which is shown in Figure 20. The decrease in sampling points causes the distortion of the original signal, leading to the occurrence of a false event. According to the results of this comparison (Figure 20), a sampling frequency of 2 Ks/s should be feasible but still needs more data to support it.

Therefore, if the proposed method is used for event detection, it is recommended that the data sampling frequency must be greater than or equal to 2 Ks/s, which could result in less distortion of the captured data.

6.4. Case Study

In order to verify whether the sampling rate (2 Ks/s) and slope threshold (0.06) for judging events can be applied in the proposed method, a mixed-load scenario is simulated. Several electrical switch cases were used to illustrate the process of the proposed event detector. Figure 21 shows the sequence of events. In this experiment, four appliances—a refrigerator, a cooker, a hairdryer, and an oven—were considered for energizing and de-energizing event detection. These electrical appliances were first activated and then deactivated in sequence. The experiment lasted approximately 90 s. The data were collected at the laboratory at the standard voltage of 120 V, a frequency of 60 Hz with a sampling frequency of 2 kHz, and then were fed to the event detector at an interval of 1.5 s with a sliding window of 3 s. Practical examples were used to demonstrate how the CUSUM geometry of RMS current was used in the proposed algorithm.

When the window length and interval were set to 3 and 1.5 s, respectively, 8 events and 41 non-events were generated within 90 s. As mentioned in Section 4.2, after every 3 s, the RMS current was converted into a CUSUM plot, and the location at which the current changed indicated a turning point on the CUSUM plot. Therefore, the slope and maximum value of the turning point were used to judge whether an event had occurred. As detailed in

Table 5. RMS current diagrams and their corresponding CUSUM diagrams for events.

Event Type	1	2	3	4
Plot of $I_{rms}$
Plot of CUSUM
Slope (stepA5)	5.46	2.79	9.28	5.11
Event type	5	6	7	8
Plot of $I_{rms}$
Plot of CUSUM
Slope (stepA5)	4.81	7.85	4.89	0.27

, the slope of the turning point is described as step A5 in Figure 9. In this case, because all slope values were greater than the 0.06 threshold, eight events were correctly identified.

Not all non-event types were present in every process of event detection. Only two types of non-events (listed in

Table 6. RMS current diagrams and their corresponding CUSUM Diagrams for non-events.

Non-Event Type	1	Non-Event Type	2
Plot of $I_{rms}$		Plot of $I_{rms}$
Plot of CUSUM		Zoom-in Plot of $I_{rms}$
Result	Slope: 0.018 < 0.06 (step A5)	Result	1.57% < 3%(step A3)

) occurred in the test case, although 41 non-events were detected. The first non-event did not pass step A5. Therefore, it was judged as a non-event, although it had passed steps A1–A4. The last non-event appeared to be in a stable state. Zooming into its graph revealed sine wave behavior. At a window length of 3 s, the maximum $I_{rms}$ was 6.8782 A, whereas the minimum $I_{rms}$ was 6.7705 A. After step A3 was used in the calculation process, the criterion of being less than 3% was not satisfied. Therefore, the event was correctly classified as a non-event.

After the event detection process was completed by the proposed event detection algorithm, 8 events and 41 non-events were successfully detected in the test case, as expected. So far, CUSUM geometric figures were successfully used as the basis for event detection.

In several event detection methods involving CUSUMs, only the judgment method of the dataset turning point has been enhanced. For example, in [11], the judgment method was changed to an adaptive method to improve the accuracy of event detection, but the mathematical method remained relatively complex. However, the event detection algorithm proposed in this study can achieve the same goal by using only a slope. In addition, for a captured current sequence, the proposed CUSUM method can be used to determine whether this sequence represents a steady state, a gradually increasing current, or a gradually decreasing current. The algorithm also provides an identification method for noise processing, thus increasing the integrity and accuracy of the event detector. Those differ from other studies that have relied on CUSUMs for event detection.

7. Conclusions

In this study, an event detection method based on the geometric features of CUSUMs was proposed. Analysis of step-like, long-transient, increasing-current, and decreasing-current events can aid in determining whether an event occurs. When an event occurs, the RMS value of the transient current can be converted into a CUSUM graph with time series, and the turning point of the graph can be used as the basis for event detection. In this study, accuracy comparisons were conducted using the COOLL and PLAID datasets. The experimental results indicated that the proposed method is more accurate than the ML method [7] and traditional CUSUM methods [11], and the method can be performed on appliances with power between 10 W and 2700 W, which is better than [7] (above 15 W) and [11] (60 W~1800 W). The window length for each event detection is also shortened to 3 s, which is faster than 5 s in [7] (not available in [11]). At the same time, a sampling frequency of 2 Ks/s and the criterion (0.06) to judge an event are also used to detect events on private electrical appliances, and a very high success detection rate was obtained. In recent years, most household appliances have been equipped with specialized electronic components designed to save energy, which increases the complexity of the current waveform. Therefore, to prevent misjudgment of event detection caused by current diversity, the detection method must be continuously improved.