Event-Triggered Transmission of Sensor Measurements Using Twin Hybrid Filters for Renewable Energy Resource Management Systems

Abstract

Recently, solar and wind power generation have gained attention as pathways to achieving carbon neutrality, and Renewable Energy Resource Management System (RERMS) technology has been developed to monitor and control small-scale, distributed renewable energy resources. In this work, we present an Event-Triggered Transmission (ETT) algorithm for RERMS, which transmits sensor measurements to the base station only when necessary. The ETT algorithm helps prevent congestion in the communication channel between RERMS and the base station, avoiding time delays or packet loss caused by the excessive transmission of sensor measurements. We design a hybrid state estimation algorithm that combines Kalman and Finite Impulse Response (FIR) filters to enhance the estimation performance, and we propose a new ETT algorithm based on this design. We evaluate the performance of the proposed algorithm through experiments that transmit actual sensor measurements from a photovoltaic power generation system to the base station, demonstrating that it outperforms existing algorithms.

1. Introduction

In recent years, there has been growing interest in renewable energy resources (RERs), such as solar and wind, to achieve carbon neutrality, leading to a rapid increase in small-scale, distributed RERs [1,2,3,4,5]. However, solar and wind energies are volatile and unpredictable, which means they may not always produce enough power to meet demand, and at times may generate more than is needed. The volatility and intermittency of RERs can hinder the stability of the power grid, necessitating a system that monitors and manages power generation and system status in real time. Recently, Renewable Energy Resource Management Systems (RERMSs) have gained attention for their ability to use information and communication technology to manage RERs [6,7,8]. RERMSs collect power generation and environmental data from RERs, visualize and display critical information, record and manage data, trigger alarms when faults occur, and analyze sensor measurements to locate and report faults.

RERMS is a technology that enables smart grids with reliable real-time information exchange between RERs and base stations at its core. To effectively monitor the state information of multiple RERs connected to the grid, it is essential to transmit this information accurately and in real time to the base station. However, communication channels have limited capacity for data transmission. When information from multiple RERs is directed to a single base station simultaneously, the volume may exceed the channel’s limit, potentially leading to transmission delays or data loss. To address this issue, event-triggered transmission (ETT) mechanisms have been developed to transmit information only when necessary, rather than at regular time intervals [9,10,11,12]. In ETT mechanisms, information is transmitted only when a specific events occur, such as when the change in sensor information exceeds a set threshold. This reduces the communication burden and energy consumption for information transmission in distributed sensor networks. In addition to RERMS, ETT algorithms have recently been used in various fields such as distributed energy storage systems in smart grids [13], the network-based control of distributed renewable energy plants [14], environmental monitoring based on sensor networks [15], and the Internet of Things (IoT) [16]. The most prevalent ETT algorithm transmits data when the discrepancy between the current sensor measurement and the most recently transmitted measurement surpasses a predefined, fixed threshold [17,18,19,20]. This is referred to as static ETT [21,22].

The transmission of data in sensor networks is subject to several potential issues, including contamination with noise, distortion from cyberattacks, and time delays caused by the presence of unnecessarily large amounts of data. Accordingly, data processing is necessary to mitigate the impact of noise and distortion, as well as to manage the volume of data, prior to transmitting sensor measurements. To this end, the state estimation algorithms (e.g., Kalman filter) can be employed to derive a more precise estimate of the underlying physical quantity from the noisy sensor measurements. Once the estimated value reaches a predefined level of accuracy, the base station server can utilize it instead of transmitting the raw measurements, thereby reducing the overall volume of transmitted data [23]. In the context of sensor networks, ETT mechanisms that employ state estimation are demonstrably more effective than their static counterparts when confronted with significant noise [24,25,26].

The Kalman filter (KF) is a well-established state estimator that has been successfully applied in numerous control and signal processing applications in recent decades. Given the state–space model of a system, the KF uses noisy measurements to estimate more accurate values of the state variables. The KF algorithm initializes the state estimate and then continuously updates it through processes known as time and measurement updates. As a result, the KF’s current estimate reflects all past information. However, if there is an error in that information, it can accumulate over time, reducing the accuracy of the estimate or even leading to filter divergence. To address this issue, estimation methods that rely on a finite window of information have been studied. This approach is referred to as stochastic finite impulse response (FIR) filtering or finite memory estimation [27,28,29,30]. Due to their finite memory structure, stochastic FIR filters are not susceptible to error accumulation, ensuring the reliability of their estimates even in the presence of modeling or computational errors. However, in situations where the error in the information is negligible, the KF, which uses all past information, is generally more accurate than the FIR filter, which uses only a subset of the information. Therefore, it is beneficial to combine the strengths of both methods by integrating the KF’s infinite memory structure with the FIR filter’s finite memory structure [31,32,33,34].

In this study, we propose a novel ETT algorithm for RERMS. Instead of transmitting raw measurements like the static ETT, we use stochastic filters (i.e., state estimation algorithms) to refine noisy measurements. In our approach, twin filters are present at both the RER and the base station. These twin filters predict the current state variable without using the current measurements. The ETT system in the RER decides whether to transmit the current measurements based on a predefined criterion. If the criterion is met, no transmission occurs, and the base station uses the predicted value for RER management. However, if the prediction does not meet the criterion, the measurement is transmitted, and the twin filters utilize the current measurement to make a more accurate estimate. We use a random process model to handle the unpredictable data from the RER. However, discrepancies between the model and the actual data can be significant, depending on the circumstances. This modeling error can lead to degradation in estimation accuracy or even filter divergence. To address this issue, we design a hybrid filter that integrates the KF and FIR filter to enhance estimation accuracy and reliability. The proposed filter, called the hybrid Kalman/FIR filter (HKFF), uses the KF as the main filter and employs a secondary FIR filter to reset and recover the main filter from failures. We propose an ETT algorithm using twin HKFFs and demonstrate its performance through comparisons with conventional ETT algorithms. Experiments were conducted to transmit active power and phase voltage measurements from a photovoltaic system to the base station. We compared static ETT, KF-based ETT, and HKFF-based ETT for transmitting measurements, and found that HKFF-ETT outperformed the other algorithms.

The remainder of this paper is organized as follows. Section 2 describes the ETT scheme using twin filters for RERMS. Section 3 derives the HKFF and proposes an HKFF-based ETT algorithm. Section 4 presents a comparative analysis of the proposed algorithm with existing conventional ETT algorithms, using experimental data. Finally, Section 5 provides the conclusions and outlines future work.

2. ETT Scheme for RERMS

In this study, we examine a photovoltaic system as a form of RER, focusing on the key physical quantities of active power and phase voltage. The values of these RER data exhibit significant variation depending on the season, time of day, and weather conditions. Since the fluctuations in these quantities are inherently unpredictable, we introduce the Gaussian noise acceleration model to represent changes in RER data. This model defines two state variables: a physical quantity, represented by x, and its rate of change over time, represented by its derivative, $\dot{x}$. The transition of these two state variables can then be expressed in discrete time as follows:

$$\begin{array}{cc}\hfill x_k& =x_{k-1}+T{\dot{x}}_{k-1}+{\displaystyle \frac{T^2}{2}}{w}_{k-1},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\dot{x}}_k& ={\dot{x}}_{k-1}+T{w}_{k-1},\hfill \end{array}$$

(2)

where T is the sampling interval, $w_{k-1}$ is the white Gaussian noise with zero-mean and $q^2$ is the variance at the discrete time step $k-1$. We define a state vector as ${\mathbf{x}}_k={\left[x_k{\dot{x}}_k\right]}^T$. The above equations can then be rewritten in matrix form as follows:

$$\begin{array}{cc}\hfill {\mathbf{x}}_k& =\mathbf{A}{\mathbf{x}}_{k-1}+\mathbf{G}{w}_{k-1},\hfill \end{array}$$

(3)

where

$$\mathbf{A}=\left[\begin{array}{cc}1& T\\ 0& 1\end{array}\right],\mathbf{G}=\left[\begin{array}{c}{T}^2/2\\ T\end{array}\right].$$

(4)

This model is also referred to as the constant velocity model because the velocity ${\dot{x}}_k$ is assumed to remain constant over the sampling interval T. In practice, however, the rate of change (velocity) of a physical quantity is not always constant. The discrepancy between the model and reality is negligible when the sampling interval is sufficiently short; however, when the interval is lengthy, the discrepancy can become significant. The process noise $w_{k-1}$, which corresponds to acceleration, is assumed to be Gaussian random noise, but it is necessary to determine the variance of this noise accurately. If the specified variance does not reflect the actual conditions, it will introduce modeling errors. Therefore, it is crucial to set the variance $q^2$ in accordance with the observed degree of variation in the physical quantity. Failing to do so may result in modeling errors, leading to inaccurate estimates produced by the filter.

To construct a state–space model, we also need a measurement equation that expresses the relationship between the measurements and the state vector. The physical quantities in photovoltaics are measured directly by sensors, and the measurement equations can be written as follows:

$$\begin{array}{cc}\hfill y_k& =\mathbf{C}{\mathbf{x}}_k+v_k,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \mathbf{C}& =\left[\begin{array}{cc}1& 0\end{array}\right].\hfill \end{array}$$

(6)

where $v_k$ represents the measurement noise, which is assumed to be zero-mean white Gaussian noise with a variance of $r^2$. The measurement noise variance $r^2$ is related to the accuracy of the sensor and can be provided by the sensor manufacturer in the manual or obtained through repeated measurements.

State estimation algorithms use state–space models and sensor measurements to estimate state variables. If the variable to be estimated is the same as the measured variable, the algorithm should provide a more accurate estimate (closer to the true value) than the measured value; otherwise, the algorithm is not worthwhile. However, state estimation algorithms can produce inaccurate results if the initial estimates or noise variance values are poorly selected. The accumulation of modeling or computational errors can lead to a divergence phenomenon, where estimates increasingly deviate from the true value. ETT algorithms based on state estimation can outperform static ETT algorithms that simply transmit measurements. However, they can also be less accurate than static ETT if the initial estimates or noise covariance values are incorrect, or if modeling or calculation errors accumulate.

The estimation process of the KF consists of two steps: time update (prediction) and measurement update (correction). The time update process does not use measurements; rather, it relies solely on the state equation to update the estimate from the previous time step to the current one. The estimate obtained from the time update process is called an a priori estimate because it is derived before the measurements are obtained (without using them). The measurement update process of the KF uses measurement equations and sensor measurements to refine state estimates. The estimate obtained from the measurement update is called an a posteriori estimate because it is calculated using the measurements (after they have been obtained). In general, the a priori and a posteriori estimates of the KF are denoted by the superscripts ‘−’ and ‘+’, respectively. The time update equations of the KF are represented as follows:

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_k^{-}& =\mathbf{A}{\widehat{\mathbf{x}}}_{k-1}^{+},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathbf{P}}_k^{-}& =\mathbf{A}{\mathbf{P}}_{k-1}^{+}{\mathbf{A}}^T+\mathbf{G}{q}^2{\mathbf{G}}^T,\hfill \end{array}$$

(8)

where $\widehat{\mathbf{x}}$ and $\mathbf{P}$ are the estimated state vector and estimation error covariance, respectively. The measurement update equations of KF are represented as follows:

$$\begin{array}{cc}\hfill {\mathbf{K}}_k& ={\mathbf{P}}_k^{-}{\mathbf{H}}^T{(\mathbf{C}{\mathbf{P}}_k^{-}{\mathbf{C}}^T+r^2)}^{-1},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_k^{+}& ={\widehat{\mathbf{x}}}_k^{-}+{\mathbf{K}}_k(y_k-\mathbf{C}{\widehat{\mathbf{x}}}_k^{-}),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\mathbf{P}}_k^{+}& =({\mathbf{I}}_2-{\mathbf{K}}_k\mathbf{C}){\mathbf{P}}_k^{-},\hfill \end{array}$$

(11)

where ${\mathbf{K}}_k$ is the Kalman gain and ${\mathbf{I}}_2$ is the 2 identity matrix.

In ETT based on the KF, a measurement is transmitted if the error of the a priori estimate, obtained without the measurement, is greater than a set value; otherwise, it is not transmitted. To achieve this, we use the Mahalanobis distance to determine the distance between a point and a distribution. If the Mahalanobis distance between the estimate and the measurement distribution exceeds a threshold, we consider the estimate unreliable and transmit the sensor measurement. The Mahalanobis distance, denoted by ${\mathcal{D}}_k$, can be calculated as follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_k& =({\widehat{x}}_k-y_k)r_k^{-1}({\widehat{x}}_k-y_k).\hfill \end{array}$$

(12)

The threshold for the Mahalanobis distance can be found in the chi-square table, where the chi-square value for a $99\%$ confidence level in a one-dimensional measurement is $6.635$. This means that when the standard deviation of the measurements is r, 99 out of 100 measurements are within a distance of $6.635$ from the mean value. In other words, measurements beyond this distance are not considered reliable compared to normal measurements. Since the $99\%$ confidence chi-square value represents the maximum range within which the estimate is reliable, using this value as a threshold will result in minimal transmissions, and the accuracy of the base station’s estimate will be very low. Therefore, the $99\%$ confidence chi-square value can be multiplied by an appropriate scale factor, $0\le \alpha \le 1$, and used as a threshold. Setting $\alpha$ to 0 means transmitting all measurements, while setting $\alpha$ to 1 means transmitting measurements at a minimum. The triggering condition for transmission can then be represented as follows:

$$\begin{array}{cc}\hfill {\eta }_k& =\left\{\begin{array}{cc}1,\hfill & {\mathcal{D}}_k>\alpha {\chi }^2,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

(13)

where ${\eta }_k$ is the triggering variable, and ${\eta }_k=1$ means that the measurement transmission is triggered. Figure 1 shows the structure of the ETT mechanism with twin filters. In this structure, KFs or HKFFs can be used in place of the twin filters.

The static ETT algorithm, which does not use state estimation, relies on the difference between the current measurement $y_k$ and the most recently transmitted value ${\overline{y}}_k$ as the error. The threshold value for the error is the magnitude of the most recently measured value multiplied by a scale factor, plus a constant value. The triggering condition for the static ETT algorithm can be expressed as follows:

$$\begin{array}{cc}\hfill {\overline{\eta }}_k& =\left\{\begin{array}{cc}1,\hfill & |y_k-{\overline{y}}_k| \ge \beta |{\overline{y}}_k| + \gamma ,\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

(14)

where the scale factor $\beta$ and the bias $\gamma$ are design parameters.

3. Proposed Algorithm

In this section, we derive a new FIR filter that has complementary properties to the KF and then design a hybrid filtering algorithm that combines the KF and the FIR filter, which we call HKFF. Finally, we propose an HKFF-based ETT algorithm.

The FIR filter is a batch processing estimation algorithm that collects a finite number of recent measurements and processes them all at once. The state variable at the current time step k is estimated using the measurements over the time horizon $[k-N,k-1]$, where N is the number of measurements used for estimation and is referred to as the memory size. The FIR filter constructs a large matrix of measurements in the interval $[k-N,k-1]$ and multiplies it by the filter gain to produce an estimate. The matrix constructed by collecting the past N measurements at time k (i.e., the measurements in the interval $[k-N,k-1]$) is denoted by ${\overline{\mathbf{Y}}}_k$ and is defined as

$$\begin{array}{cc}\hfill {\overline{\mathbf{Y}}}_k& ={[y_{k-N}{y}_{k-N+1}\cdots y_{k-1}]}^T.\hfill \end{array}$$

(15)

Similarly, we define the matrices that collect the process and measurement noise in the interval $[k-N,k-1]$ as follows:

$$\begin{array}{cc}\hfill {\overline{\mathbf{W}}}_k& ={[w_{k-N}{w}_{k-N+1}\cdots w_{k-1}]}^T,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\overline{\mathbf{V}}}_k& ={[v_{k-N}{v}_{k-N+1}\cdots v_{k-1}]}^T.\hfill \end{array}$$

(17)

Using the state–space models (3) and (5), a functional relationship between ${\overline{\mathbf{Y}}}_k$ and the current state ${\mathbf{x}}_k$ can be expressed as follows:

$$\begin{array}{cc}\hfill {\overline{\mathbf{Y}}}_k& ={\overline{\mathbf{C}}}_k{\mathbf{x}}_k+{\overline{\mathbf{G}}}_k{\overline{\mathbf{W}}}_k+{\overline{\mathbf{V}}}_k,\hfill \end{array}$$

(18)

where

$$\begin{array}{cc}\hfill {\overline{\mathbf{C}}}_k& =\left[\begin{array}{c}\mathbf{C}{\mathcal{F}}_N\\ \mathbf{C}{\mathcal{F}}_{N-1}\\ ⋮\\ \mathbf{C}{\mathcal{F}}_1\end{array}\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\overline{\mathbf{G}}}_k& =-\left[\begin{array}{cccc}\mathbf{C}{\mathcal{F}}_1\mathbf{G}& \mathbf{C}{\mathcal{F}}_2\mathbf{G}& \cdots & \mathbf{C}{\mathcal{F}}_1\mathbf{G}\\ 0& \mathbf{C}{\mathcal{F}}_1\mathbf{G}& \cdots & \mathbf{C}{\mathcal{F}}_{N-1}\mathbf{G}\\ ⋮& ⋮& ⋮& ⋮\\ 0& 0& \cdots & \mathbf{C}{\mathcal{F}}_1\mathbf{G}\end{array}\right],\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathcal{F}}_n& =\left[\begin{array}{cc}1& -d_{n}T\\ 0& 1\end{array}\right],\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill d_n& =t_1-t_n,\hfill \end{array}$$

(22)

where $t_1$ is the time point of most recent measurement transmission, and $t_n$ is the n-th time point of transmission going backward from the most recent.

We rewrite the Equation (18) as follows:

$$\begin{array}{cc}\hfill {\overline{\mathbf{Y}}}_k-{\overline{\mathbf{C}}}_k{\mathbf{x}}_k& ={\overline{\mathbf{G}}}_k{\overline{\mathbf{W}}}_k+{\overline{\mathbf{V}}}_k.\hfill \end{array}$$

(23)

In (23), ${\overline{\mathbf{C}}}_k{\mathbf{x}}_k$ is the predicted output obtained by multiplying the measurement matrix by the state vector, and ${\overline{\mathbf{Y}}}_k$ is the measured output. Thus, the left-hand side of (23) corresponds to the output error. We will derive an FIR filter that minimizes the square of the output error as follows:

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_k& =\arg \underset{{\mathbf{x}}_k}{\min }{({\overline{\mathbf{Y}}}_k-{\overline{\mathbf{C}}}_k{\mathbf{x}}_k)}^T({\overline{\mathbf{Y}}}_k-{\overline{\mathbf{C}}}_k{\mathbf{x}}_k).\hfill \end{array}$$

(24)

Expanding the square of the output error, we obtain

$${({\overline{\mathbf{Y}}}_k-{\overline{\mathbf{C}}}_k{\mathbf{x}}_k)}^T({\overline{\mathbf{Y}}}_k-{\overline{\mathbf{C}}}_k{\mathbf{x}}_k)={\overline{\mathbf{Y}}}_k^T{\overline{\mathbf{Y}}}_k-{\overline{\mathbf{Y}}}_k^T{\overline{\mathbf{C}}}_k{\mathbf{x}}_k-{\mathbf{x}}_k^T{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{Y}}}_k+{\mathbf{x}}_k^T{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k{\mathbf{x}}_k.$$

(25)

The partial derivative of (25) with respect to ${\mathbf{x}}_k$ is

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial ({\overline{\mathbf{Y}}}_k^T{\overline{\mathbf{Y}}}_k-{\overline{\mathbf{Y}}}_k^T{\overline{\mathbf{C}}}_k{\mathbf{x}}_k-{\mathbf{x}}_k^T{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{Y}}}_k+{\mathbf{x}}_k^T{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k{\mathbf{x}}_k)}{\partial {\mathbf{x}}_k}}=-2{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{Y}}}_k+2{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k{\mathbf{x}}_k.\end{array}$$

(26)

Finding ${\mathbf{x}}_k$ that makes the above expression equal to zero, we obtain ${\widehat{\mathbf{x}}}_k$ as follows:

$${\widehat{\mathbf{x}}}_k={\left({\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k\right)}^{-1}{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{Y}}}_k.$$

(27)

We will prove that the estimate of the FIR filter (27) is an unbiased estimate. Substituting (18) for ${\overline{\mathbf{Y}}}_k$ in (27), we obtain

$${\widehat{\mathbf{x}}}_k={\left({\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k\right)}^{-1}{\overline{\mathbf{C}}}_k^T({\overline{\mathbf{C}}}_k{\mathbf{x}}_k+{\overline{\mathbf{G}}}_k{\overline{\mathbf{W}}}_k+{\overline{\mathbf{V}}}_k).$$

(28)

Taking expectations on both sides of (28) gives

$$E\left[{\widehat{\mathbf{x}}}_k\right]=E\left[{\left({\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k\right)}^{-1}{\overline{\mathbf{C}}}_k^T({\overline{\mathbf{C}}}_k{\mathbf{x}}_k+{\overline{\mathbf{G}}}_k{\overline{\mathbf{W}}}_k+{\overline{\mathbf{V}}}_k)\right]$$

(29)

$$\begin{array}{cc}\hfill & ={\left({\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k\right)}^{-1}{\overline{\mathbf{C}}}_k^{T}E[{\overline{\mathbf{C}}}_k{\mathbf{x}}_k+{\overline{\mathbf{G}}}_k{\overline{\mathbf{W}}}_k+{\overline{\mathbf{V}}}_k]\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & ={\left({\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_k\right)}^{-1}{\overline{\mathbf{C}}}_k^T{\overline{\mathbf{C}}}_{k}E\left[{\mathbf{x}}_k\right]\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & =E\left[{\mathbf{x}}_k\right],\hfill \end{array}$$

(32)

where the expectations for the noise terms ${\overline{\mathbf{W}}}_k$ and ${\overline{\mathbf{V}}}_k$ are zero because we assumed zero-mean Gaussian noise. Since the expectation of the estimate is equal to the expectation of the true value (i.e., $E\left[{\widehat{\mathbf{x}}}_k\right]=E\left[{\mathbf{x}}_k\right]$), the FIR filter is an unbiased estimator.

Now, we design a hybrid estimation algorithm that combines the KF and the FIR filter. There are three approaches to combining the two different types of estimation algorithms: (1) fusion, where the estimates of the two algorithms are appropriately weighted to obtain a weighted sum; (2) switching, where the more reliable of the two algorithms is chosen depending on the situation; and (3) complementing the main algorithm with a secondary algorithm. We take the third approach and design a hybrid algorithm using the KF as the main algorithm and the FIR filter as the secondary algorithm. This is because the KF is more accurate and less computationally intensive than the FIR filter in normal situations. In our proposed hybrid estimation algorithm, the FIR filter only works in situations where the KF becomes very inaccurate due to modeling or computational errors and resets the KF to the FIR filter’s estimate. The FIR filter is a batch-processing algorithm and can produce an estimate given only a set of measurements. Thus, the FIR filter is suitable as a secondary filter that runs only when needed to gain information for resetting. The KF performs a recursive update and accumulates past information, so it is highly accurate under normal conditions. However, if there are significant modeling errors, which can accumulate over time and cause the filter to diverge, the FIR filter does not accumulate information, giving it the advantage of not accumulating error, but it is less accurate than the KF under normal conditions. This complementary nature is similar to the relationship between Global Navigation Satellite Systems (GNSSs) and Inertial Navigation Systems (INSs). Thus, we propose to use the FIR filter to reset the KF, similarly to how GNSS is used to reset INS.

We use the Mahalanobis distance between the filter’s estimate and the sensor measurement as a metric to evaluate estimation accuracy. If the Mahalanobis distance of the KF is much larger than that of the FIR filter, we reset the KF using the FIR filter’s output. The reset condition adopting a scaling factor is described as follows:

$$\begin{array}{cc}\hfill {\eta }_{\mathrm{reset}}& =\left\{\begin{array}{cc}1,\hfill & {\mathcal{D}}_{\mathrm{KF}}>a{\mathcal{D}}_{\mathrm{FIR}},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

(33)

where ${\eta }_{\mathrm{reset}}$ is a variable for triggering the reset of KF by the FIR filter; ${\mathcal{D}}_{\mathrm{KF}}$ and ${\mathcal{D}}_{\mathrm{FIR}}$ are the Mahalanobis distances of the KF and FIR filter, respectively; and $0\le a\le 1$ is the scale factor.

In the HKFF algorithm, it is the scaling factor a that determines the switching between the FIR filter and the KF. If the value of a is set to $0.5$, it means that the KF switches to the FIR filter when the Mahalanobis distance of the KF is more than twice that of the FIR filter. The reason for the scaling factor is that the Mahalanobis distance is not an exact reflection of the estimation error of each filter, so if the Mahalanobis distance of the KF is higher than that of the FIR filter, it does not mean that the KF’s estimation is more inaccurate than the FIR filter. However, if the Mahalanobis distance value of the KF is much higher than that of the FIR filter, then we can conclude that the KF estimate is more inaccurate than the FIR filter. The comparison criterion is determined by the scale factor a, which can only be determined by the engineer’s experience. The existing ETT algorithm also requires a threshold to determine the triggering, which is also determined by the engineer’s experience without a systematic selection method. In the HKFF algorithm, the value of the scale factor must be determined after sufficient simulation and experimentation of the target system and analysis of its characteristics.

The use of HKFF instead of KF in the ETT algorithm, with twin filters described in the previous section results in an HKFF-based ETT algorithm, is summarized in the pseudocode in Algorithm 1.

Algorithm 1 HKFF-based ETT

1: for $k=1,2,\cdots$ do 2:      Obtain a priori estimate of KF ${\widehat{\mathbf{x}}}_k^{-}$ using Equation (7). 3:      Compute the Mahalanobis distance ${\mathcal{D}}_{\mathrm{KF}}$ for ${\widehat{\mathbf{x}}}_k^{-}$ using Equation (12). 4:      if ${\mathcal{D}}_{\mathrm{KF}}>{\chi }^2$ then 5:           ${\eta }_k=1$ 6:           Obtain a posteriori estimate of KF ${\widehat{\mathbf{x}}}_k^{+}$ using Equation (9). 7:      else 8:           ${\eta }_k=0$ 9:           ${\widehat{\mathbf{x}}}_k^{+}\leftarrow {\widehat{\mathbf{x}}}_k^{-}$ 10:      end if 11:      if $k>N$ then 12:           Obtain estimate of FIR filter ${\widehat{\mathbf{x}}}_k$ using Equation (27). 13:           Compute the Mahalanobis distance ${\mathcal{D}}_{\mathrm{FIR}}$ for ${\widehat{\mathbf{x}}}_k$ using Equation (12). 14:           if ${\mathcal{D}}_{\mathrm{FIR}}<a{\mathcal{D}}_{\mathrm{KF}}$ then 15:              ${\widehat{\mathbf{x}}}_k^{+}\leftarrow {\widehat{\mathbf{x}}}_k$ 16:           end if 17:      end if 18: end for

In this manuscript, the AI tool (ChatGPT 4.0) was used solely to check for grammatical errors in English sentences.

4. Experiments

In this section, we present experimental results to validate the performance of the proposed HKFF-ETT algorithm. The experimental data were obtained from a photovoltaic system installed on the roof of the Korea Electrotechnology Research Institute. The power generation system, shown in Figure 2a, can adjust the tilt and orientation angles of the photovoltaic panels using motors. The photovoltaic system is equipped with a variety of sensors to measure physical quantities related to power generation, such as solar radiation and panel temperature, as well as the active power produced by the photovoltaic panels and the voltage. Sensor measurements are transmitted to the server computer in the base station via Ethernet communication with the data transmission device, as shown in Figure 2b. The solar power data exhibit a repeating pattern of variation with a period of one day, as shown in Figure 2c. We measured the active power generated and the phase voltage at one-minute intervals to obtain one day of data.

The ETT algorithm must transmit as little data as possible while being as accurate as possible. Therefore, to evaluate the performance of the ETT algorithm, we need to calculate the accuracy of the transmitted data and the transmission rate. The transmission rate can be easily obtained by dividing the number of transmissions by the total number of data samples. However, measuring the accuracy of the transmitted data requires more consideration. State estimation algorithms use noisy sensor measurements to estimate the true value of a state variable. Therefore, the true values are required to calculate the estimation error. ETT algorithms based on state estimation algorithms, such as KF and HKFF, produce estimates and use them instead of measurements, so they require the true values of physical quantities to measure the accuracy of the data. However, the actual values of the physical quantities are unknown to us, and all we have are the sensor readings. Therefore, we used the measurements from the sensors as a substitute for the true values and artificially added noise to them as new sensor measurements. The new sensor measurement with artificial noise was processed by the ETT algorithm, and the error was obtained by comparing the transmitted (or estimated) value with the original value without the added noise. We ran 1000 Monte Carlo (MC) simulations with randomly generated noise and computed the Root-Mean-Square Error (RMSE) defined as follows:

$$\begin{array}{c}\hfill e\left(k\right)=\sqrt{{\displaystyle \frac{1}{1000}}\sum _{i=1}^{1000}{(x_k^i-{\widehat{x}}_k^i)}^2},\end{array}$$

(34)

where $x_k$ and ${\widehat{x}}_k$ are the true and estimated values of the sensor data obtained from the i-th MC run, respectively. The RMSE represents the error at each time step, and we further evaluated the overall accuracy of the ETT algorithm by computing the time-averaged RMSE (TARMSE), which is defined as

$$\begin{array}{c}\hfill \mathrm{TARMSE}={\displaystyle \frac{1}{1440}}\sum _{k=1}^{1440}e\left(k\right),\end{array}$$

(35)

where 1440 is the total number of data samples in a 24 h period measured at 1 min intervals. However, the value of TARMSE depends on the type of measurement and the amount of noise. To better compare the accuracy of the algorithms, we devised a new performance metric called the accuracy ratio, which is defined as follows:

$$\begin{array}{cc}\hfill \mathrm{Accuracy}\mathrm{Ratio}& ={\displaystyle \frac{\mathrm{Accuracy}\mathrm{of}\mathrm{ETT}\mathrm{Algorithm}}{\mathrm{Accuracy}\mathrm{of}\mathrm{Measurement}}}\times 100,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill \mathrm{Accuracy}& ={\displaystyle \frac{1}{\mathrm{TARMSE}}}.\hfill \end{array}$$

(37)

In addition, the transmission rate is defined as

$$\begin{array}{cc}\hfill \mathrm{Transmission}\mathrm{Rate}& ={\displaystyle \frac{\mathrm{Number}\mathrm{of}\mathrm{Transmission}}{\mathrm{Number}\mathrm{of}\mathrm{Samples}}}\times 100.\hfill \end{array}$$

(38)

The units for the accuracy ratio and transmission rate are in percentage. Lastly, we devised a new performance metric called the score, which is defined as

$$\begin{array}{cc}\hfill \mathrm{Score}& ={\displaystyle \frac{\mathrm{Accuracy}\mathrm{Rate}}{\mathrm{Transmission}\mathrm{Rate}}}.\hfill \end{array}$$

(39)

The score is used to compare the performances of two algorithms by calculating the accuracy ratio per $1\%$ of the transmission rate when the accuracy and transmission rates of the two algorithms are similar, making it difficult to compare their performances. The score has no units. We applied three ETT algorithms—KF-ETT, HKFF-ETT, and static ETT—to two types of measured data: active power and phase voltage.

4.1. ETT of Active Power Measurements

First, we applied the ETT algorithms to the active power measurements (in kW) produced by the photovoltaic panel over a duration of 1 s. The actual measurements were used as the true values, and Gaussian noise with a mean of 0 and a variance of $0.{05}^2$ was generated and used as the measurements. Figure 3 shows the true and measured active power, along with its estimation by the ETT algorithms. The solar power generation is zero in the absence of sunlight; however, the measured value is not zero in the absence of sunlight in Figure 3. This is because we generated artificial noise to evaluate the performance of the algorithms. We can see that the power generation gradually increases as the sun rises and then fluctuates near the peak, which is the result of a sharp drop in power generation as the sun is blocked by clouds, followed by a rise in power generation as the clouds clear. The solar power generation is characterized by a nearly constant value during the sunless hours, followed by rapid changes during the sunlit hours. When transmitting measurement data with this characteristic to the base station, it is necessary to transmit very little during the sunless hours and then transmit at a very high frequency when the power generation changes rapidly after the sun rises.

Figure 4 shows the transmission timing charts for the three ETT algorithms. In this figure, ‘1’ indicates that transmission was performed, and ‘0’ indicates that transmission was not performed. A denser distribution of vertical bars indicates a higher frequency of transmissions. It can be seen that the proposed algorithm, HKFF-ETT, transmitted very few times during periods of little power fluctuation. On the other hand, KF-ETT has the highest number of transmissions among the three algorithms, and it can be seen that it transmitted consistently even during times of little power fluctuation. Finally, static-ETT showed a very low transmission frequency during periods with little change in the measurements, but it performed relatively more transmissions compared to the proposed HKFF-ETT.

Figure 5 shows the RMSE between the values estimated (or sent) by the ETT algorithms and the true values. In this figure, a few RMSE values of the HKFF-ETT were very large, which made the width of the vertical axis of the graph excessively wide on a linear scale. Therefore, we changed the scale of the vertical axis to a logarithmic scale to observe the detailed changes of the three algorithms. HKFF-ETT was very effective when the measurements were constant with little variation. In Figure 5, the HKFF-ETT shows much lower RMSE values than the measurements in the period when solar power is almost zero. This is not surprising because state estimation algorithms utilize noisy measurements to obtain more accurate state estimates than the measurements themselves. However, in this case, due to the low accuracy of the model, HKFF has a higher estimation accuracy in the low-variation regime but exhibits the highest RMSE of the three algorithms in the high-variation regime. It is important to consider that the high RMSE in that region is also the result of transmitting at a very low frequency. KF-ETT exhibited a constant RMSE without much variation, and its value is slightly higher than the RMSE of the measurements. However, this high accuracy and consistent performance result from transmitting measurements at a consistently high frequency. KF-ETT also shows an increase in RMSE of around 750 min when the power generation drops sharply. Since the static ETT algorithm does not estimate physical quantities but simply transmits measurements intermittently, it cannot be more accurate than the measurements. Therefore, the RMSE of static ETT cannot be lower than the RMSE of the measurements. The RMSE of static ETT was similar to the RMSE of the measurements during periods of little fluctuation and higher than the measurements during periods of high fluctuation.

Table 1. Comprehensive performance comparison of ETT algorithms for active power measurements.

	Static ETT	KF-ETT	HKFF-ETT (Proposed)
Accuracy Ratio (%)	90.1	91.8	92.5
Transmission Rate (%)	9.9	56.7	9.0
Score	9.1	1.6	10.3

provides a comprehensive performance comparison of the ETT algorithms. The accuracy ratios for static ETT, KF-ETT, and HKFF-ETT were $90.1\%$, $91.8\%$, and $92.5\%$, respectively, because we tuned the threshold values of the three algorithms to achieve similar accuracy ratios. We compared the transmission rates when the three algorithms had similar accuracies of above $90\%$. KF-ETT had a transmission rate of $56.7\%$, which means that, out of 1440 total measurements, 817 measurements were transmitted. The static ETT and HKFF-ETT had transmission rates of $9.9\%$ and $9.0\%$, respectively. Since the accuracy and transmission rates of static ETT and HKFF-ETT are similar, we can use the score to compare their performance. The calculated score value is 9.1 for static ETT and 10.3 for HKFF-ETT, indicating that the proposed HKFF-ETT performs better.

4.2. ETT of Phase Voltage Measurements

Second, the ETT algorithms were applied to the transmission of phase voltage measurements. The photovoltaic system originally generates direct current, but it is converted to a three-phase alternating current by an inverter to connect to the grid, and the phase voltage is maintained in the range of $\pm 6\%$ at 220 V. Figure 6 compares the true, measured, and estimated values of the phase voltage. However, it is difficult to see the difference between the three ETT algorithms in this figure. Figure 7 compares the transmission timing of the three ETT algorithms. Unlike the case of the active power measurements, the phase voltage changed constantly, so the transmission rate did not decrease during certain intervals, and we can see that static ETT has a higher transmission rate compared to KF-ETT and HKFF-ETT. Figure 8 shows the RMSE of the measurements and ETT algorithms. However, since the ETT algorithms were tuned to have a similar accuracy and the measurements were constantly changing, there was no variation in RMSE during any specific interval. It is difficult to compare the performance of the ETT algorithms using Figure 6 and Figure 8, but

Table 2. Comprehensive performance comparison of ETT algorithms for phase voltage measurements.

	Static ETT	KF-ETT	HKFF-ETT (Proposed)
Accuracy ratio (%)	99.3	100.2	99.5
Transmission rate (%)	64.3	34.2	30.9
Score	1.5	2.9	3.2

allows us to do so. Table 2 shows that the accuracy ratios of the three algorithms are similar because we tuned the threshold values of the ETT algorithms. Since KF is an estimation algorithm that uses noisy measurements to produce more accurate values, it is not surprising that KF-ETT has an accuracy ratio greater than $100\%$. Since the performance of KF-ETT and HKFF-ETT was similar, we can use the score for an accurate performance comparison. The score of the proposed algorithm, HKFF-ETT, was $3.2$, which is higher than $2.9$ for KF-ETT. Therefore, HKFF-ETT outperformed KF-ETT.

Figure 9 compares the performances of the ETT algorithms in the experiments for the two cases. The two cases exhibited different characteristics in sensor measurements. In the first case, the active power measurements demonstrated little change during the night and a very large variation in power values during the day. In the second case, the phase voltage measurements exhibited constant variation around 220 V. For the two cases with such different characteristics, static ETT and KF-ETT performed differently. The static ETT outperformed the KF-ETT in the first case, while the KF-ETT outperformed the static ETT in the second case. Therefore, it can be seen that static ETT and KF-ETT have a large variation in performance depending on the characteristics of the data. However, the proposed HKFF-ETT showed the best performance in both cases regardless of the data characteristics.

5. Conclusions

In this study, we proposed an ETT technique for RERMS and analyzed its performance through experiments. In the proposed algorithm, the state of the RER is estimated by twin filters present at both the RER and the base station, and measurements are sent only if the reliability of the estimated value does not reach a set level. We designed a new state estimation algorithm, called HKFF, for the use of the twin filters. The active power and phase voltage were measured for one day from the photovoltaic system, and the proposed ETT algorithm was applied to the transmission of the measurements. We compared the proposed HKFF-based ETT with the existing algorithms, static ETT and KF-based ETT. ETT algorithms are less accurate when the threshold is increased because they transmit less and more accurately when the threshold is decreased because they transmit more. Therefore, we adjusted the triggering threshold for each of the three ETT algorithms so that their accuracy was similar and then compared their transmission rates. Consequently, the proposed algorithm achieved the lowest transmission rate for both experiments. The proposed algorithm achieved $90\%$ accuracy of the measurements in the active power case while sending only $9\%$ of the total measurements and $99.5\%$ accuracy of the measurements in the phase voltage case while sending only $30.9\%$ of the measurements. Therefore, we verified that the proposed algorithm can reduce the amount of communication for measurement transmission more effectively compared to the existing algorithms. However, the optimal performance of the proposed ETT can only be achieved by effectively tuning the design parameters, which is still a trial-and-error process and needs to be addressed. Therefore, in the future, we plan to study a new ETT algorithm that simplifies the design parameters and makes it easier to set up. The developed ETT algorithm is expected to be utilized in various sensor network-based systems, including RERMS, smart grid, IoT, and environmental monitoring systems.