OOK power model based dynamic error testing for smart electricity meter

With the rapid construction of smart grids in the world, smart meters are widely used in electric power systems. Smart grids show two obvious characteristics in energy generation and consumption: one is unsteady features which are generated by the wind energy and solar energy, and the other is dynamic loads which produced in electric train, steel smelting arc furnace, welding equipment etc [1–3]. Dynamic errors caused by fast active power fluctuation would occur while Smart Electricity Meters are used for metering electrical energy consumption of the dynamic loads. Therefore, for smart meters, a good dynamic error feature is required to ensure correct electric energy metering. Moreover the dynamic error testing method for smart meters in the unsteady power situation is becoming a new challenge for the researchers in the field [4].

During the past decades, there has been a significant progress on the meter's metrological characterization. Some researchers have studied the accuracy of meters in real-world unbalanced harmonic voltage and current conditions [5], whilst others analyzed the aspects related to the metrological characterization of meters in the presence of harmonic distortion or under non-sinusoidal conditions [6–10]. The most suitable waveforms have been analyzed for calibrating electrical instrument [11]. And the comparison between different methods for harmonic pollution metering has also been made [12]. Particularly, the measurement of time-dependent harmonics in modern power supply systems also attracts attention [13]. It should be pointed out that, so far, almost all concerning meters' error testing problems have been specifically considered in stationary situation with or without harmonics. As dynamic loads universally exist in electric power systems, the past few decades have witnessed significant progress on dynamic load models and vast literatures have concentrated on the topic of load active and reactive power model [14–19]. In particular, the dynamic load power model has been considered to improve calculating accuracy for its popular applications in the power system. Unfortunately, when smart meters' error testing and dynamic load come together, the problem has become quite complex because the existing dynamic load power model cannot be simply adopted and these models are not suitable for the error testing of smart meters.

As elaborated above, the dynamic error testing of smart meters has become a focus of concern for researchers during the past few years. A dynamic current waveform with a sine wave envelope used to check the dynamic responses of the meter has been proposed in literature [20].

In this work, we probed into the error testing method of smart meters' electric energy metering involving different dynamic power modes. We proposed sufficient dynamic power modes based on OOK testing dynamic current and energy models. Then we derived a novel measure algorithm for dynamic error testing, which makes the dynamic load energy measurement traceable to static electric energy standard. At last, the proposed testing system was used to measure dynamic errors of smart meters from different manufacturers and the test results are given.

The model designed for the dynamic error testing should be represented by the following four characteristics: (1) the model can provide varieties of typical dynamic power modes; (2) testing signal can be generated and controlled easily by testing equipment; (3) testing signal should be periodic in order to repeat dynamic error testing and compare testing results and based on the model, the dynamic electric energy measurement can be traceable to national electric energy standard. Thus, the investigation of the dynamic testing current, energy models and relevant dynamic error testing method for meters can help to find out whether there is a problem with the accuracy of meters under dynamic load power conditions [21].

2.1. The model of dynamic testing voltage and current

In the dynamic load conditions, the instantaneous testing voltage and testing current for dynamic errors can be expressed as follows:

$$\begin{eqnarray}&&u(t)=\sqrt{2}{{U}_{\text{rms}}}\left[{{v}_{1}}(t)\right]\sin \omega t\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&i(t)=\sqrt{2}{{I}_{\text{rms}}}\left[{{v}_{2}}(t)\right]\sin \left[\omega t-\varphi (t)\right]\end{eqnarray} \tag{ 2 }$$

Where ${{v}_{1}}(t)$ is the amplitude of the instantaneous testing voltage, ${{v}_{2}}(t)$ is the amplitude of the instantaneous testing current, $f$ is frequency, $||{{v}_{1}}(t)|{{|}_{\infty}}\leqslant 1$, $||{{v}_{2}}(t)|{{|}_{\infty}}\leqslant 1$, $\omega =2\pi f$, and $f=1/T=50\,\text{or}\,60$ Hz, $T$ is period.

Considering the condition in the realistic situations of power systems, the amplitude change of testing voltage is far smaller than that of the load testing current, and both phase and frequency of testing current change gradually between every neighbor period. In order to characterizing the condition, we define ${{v}_{1}}(t)=1$ in equation (1) and ${{v}_{2}}(t)$ as deterministic time-vary function in equation (2). In time domain, by truncating testing voltage and current, we obtain the instantaneous testing voltage ${{u}_{n}}(t)$ and current ${{i}_{n}}(t)$ function sequences in any integer period as follows,

$$\begin{eqnarray}&&{{u}_{n}}(t)=\sqrt{2}{{U}_{\text{rms}}}\left(\sin \omega t\right)\centerdot g(t-nT)\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{{i}_{n}}(t)=\sqrt{2}{{I}_{\text{rms}}}{{v}_{2}}(t)\sin \left(\omega t-{{\varphi}_{n}}\right)\centerdot g(t-nT)\end{eqnarray} \tag{ 4 }$$

Where $g(t-nT)=\left\{\begin{array}{*{35}{l}} 1\,\,nT\leqslant t\leqslant (n+1)T \\ 0\quad \text{else} \end{array}\,,\,n\in \right.$ $N$ ($N$ is the set of natural numbers) and ${{\varphi}_{n}}$ is the initial phase in the nth period.

Using the above method, the testing instantaneous voltage and current can be expressed as function sequences $\left\{{{u}_{1}}(t),{{u}_{2}}(t),\cdots,{{u}_{n}}(t)\right\}$ and $\left\{{{i}_{1}}(t),\cdots,{{i}_{n}}(t)\right\}$. Let us define an OOK (on-off-keying) control signal

$$\begin{eqnarray}&&{{a}_{n}}={{v}_{2}}(t)=\left\{\begin{array}{*{35}{l}} 1,\,\,nT\leqslant t\leqslant (n+1)T, \\ \,\,\,\,\,\,\text{signify} ~\text{the} ~\text{current} ~\text{signal} ~\text{is} ~\text{ON} ~ \\ \text{0},\,\,\text{else},\, \\ \,\,\,\,\,\,\text{signify} ~\text{the} ~\text{current} ~\text{signal} ~\text{is} ~\text{OFF} \end{array}\right. \end{eqnarray}$$

Then the truncated testing current

$$\begin{eqnarray}&&{{i}_{n}}(t)=\sqrt{2}{{I}_{\text{rms}}}{{a}_{n}}\sin \left[\omega t-{{\varphi}_{n}}\right]\centerdot g(t-nT)\end{eqnarray} \tag{ 5 }$$

Hence the instantaneous testing current can be rewritten consequently as the sum of ${{i}_{n}}(t)$:

$$\begin{eqnarray}&&i(t)=\underset{n}{\sum}\,{{i}_{n}}(t)=\underset{n}{\sum}\,\sqrt{2}{{I}_{\text{rms}}}{{a}_{n}}\sin \left[\omega t-{{\varphi}_{n}}\right]\centerdot g(t-nT)\end{eqnarray} \tag{ 6 }$$

In this work, truncated instantaneous testing current ${{i}_{n}}(t)$ was generated by the OOK control signal ${{a}_{n}}$ which modulates a steady AC current ${{i}_{\text{S}}}(t)$ from standard power source output. Equations (3)–(6) are called the model of testing voltage and OOK testing dynamic (TD) current respectively.

Figure 1 shows the waveforms of testing voltage and OOK TD current, where ${{M}_{1}}$ represents the number of period when the OOK TD current is ON (${{a}_{n}}=1$), ${{M}_{2}}$ represents the number of period when the TD current is OFF (${{a}_{n}}=0$) and the sum ${{M}_{1}}+{{M}_{2}}$ means the OOK modulation period.

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2.2. The model of OOK dynamic testing power

Based on the definition, the truncated instantaneous power can be expressed in any period as follows: ${{P}_{n}}(t)$ [22, 23],

$$\begin{eqnarray}&&{{P}_{n}}(t)={{u}_{n}}(t)\centerdot {{i}_{n}}(t)\end{eqnarray} \tag{ 7 }$$

Substituting equation (3) and (5) into the formula (7), we can obtain OOK testing dynamic power as following which is shown in figure 1.

$$\begin{eqnarray}&&{{P}_{n}}(t)={{U}_{\text{rms}}}{{I}_{\text{rms}}}{{a}_{n}}\left[\cos {{\varphi}_{n}}-\cos \left(2\omega t-{{\varphi}_{n}}\right)\right]\end{eqnarray} \tag{ 8 }$$

Therefore, the dynamic load energy drives the tested smart meter in nth period T is as follows,

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{E}_{I}}(n)=\langle {{u}_{n}}(t),{{i}_{n}}(t)\rangle T={\int}_{nT}^{(n+1)T}{{P}_{n}}(t)\text{d}t \\ \,\,\,\,\,\,\,\,\,={{U}_{\text{rms}}}{{I}_{\text{rms}}}{{a}_{n}}T\cos {{\varphi}_{n}} \end{array}\end{eqnarray} \tag{ 9 }$$

Figure 2 shows the waveforms of instantaneous testing voltage, three-phase OOK TD currents and three-phase total OOK TD power in the condition of power factor $\cos {{\varphi}_{n}}=1.0$, ${{M}_{1}}=1$ and ${{M}_{2}}=2$.

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In equation (9), let the testing root mean square voltage ${{U}_{\text{rms}}}=\alpha {{U}_{N}}$ and current ${{I}_{\text{rms}}}=\beta {{I}_{N}}$, where ${{U}_{N}}$ and ${{I}_{N}}$ are the rated voltage and current of the tested meter respectively, and let power factor $\cos {{\varphi}_{n}}={{\gamma}_{n}}$, then the model of OOK testing dynamic load energy (TDLE) can be established as

$$\begin{eqnarray}\begin{array}{l} {{E}_{I}}(n)&=\left[{{a}_{n}}\alpha \beta {{U}_{N}}{{I}_{N}}T{{\gamma}_{n}}\right] \\ & ={{a}_{n}}\left[\alpha \beta {{\gamma}_{n}}T\right]{{E}_{R}}\,\,\,\,\, ~\text{:} ~ \,n=1,2,\cdots \\ & ={{a}_{n}}{{E}_{q}} \\ \quad {{E}_{R}}&={{U}_{N}}{{I}_{N}}T\,;\,\,\,{{E}_{q}}=\left[\alpha \beta {{\gamma}_{n}}\right]{{E}_{R}} \end{array}\end{eqnarray} \tag{ 10 }$$

Where $\alpha$, $\beta$, ${{\gamma}_{n}}$ and ${{a}_{n}}$ determine different test conditions, ${{E}_{R}}$ is defined as rated TDLE quantity and ${{E}_{q}}$ is the TDLE intensity. Equation (10) shows that ${{E}_{I}}(n)$ is discrete TDLE sequence:

$$\begin{eqnarray}&&\left\{{{E}_{I}}(1),{{E}_{I}}(2),\cdots,{{E}_{I}}(n),{{E}_{I}}(n)={{a}_{n}}\centerdot {{E}_{q}}\right\}\end{eqnarray} \tag{ 11 }$$

Under OOK TDLE sequences driving, smart meter samples discrete voltage ${{u}_{n}}\left({{t}_{i}}\right)$ and current ${{i}_{n}}\left({{t}_{i}}\right)$ with time interval ${{t}_{\text{s}}}$ in every period T.

2.3. The types of TDLE sequences and the mode of OOK TD power

In order to test the dynamic errors of smart meters, two types of deterministic testing energy sequence $\left\{{{E}_{I}}(n):n=0,1,2\cdots \right\}$ are proposed in this work. One is a periodic unit sampling TDLE sequence and the other is a periodic rectangular TDLE sequence.

The period unit sampling TDLE sequence is denoted as equation (12), and the sequence intensity is ${{E}_{q}}$. The TDLE sequence is suitable to evaluate the influence of fast impact power to dynamic error of meters.

$$\begin{eqnarray}&&{{E}_{I}}(n)={{a}_{n}}\centerdot {{E}_{q}};\\ &&\,\,\,\,\,\,{{a}_{n}}=\left\{\begin{array}{*{35}{l}} 1\,\,n=LM,\,L\in N \\ 0\,\,\text{else} ~ \,M\in {{Z}^{+}}\,\left({{Z}^{+}}\,\text{is}\,\text{the}\,\text{set}\,\text{of}\,\text{positive}\,\text{integer}\right) \end{array}\right. \end{eqnarray} \tag{ 12 }$$

Figure 3 shows the physical meaning of the sequence which corresponds to the integral quantity of the TD power in each period.

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The discrete rectangular TDLE sequence is denoted as equation (13), the sequence intensity is also ${{E}_{q}}$. The TDLE sequence is suitable to estimate the influence of mid-speed impact and low speed fluctuating power to dynamic error of meter.

$$\begin{eqnarray}&&{{E}_{I}}(n)={{a}_{n}}\centerdot {{E}_{q}};\,\,\left(n=0,\ldots,\,N-1\right)\end{eqnarray}$$

$$\begin{eqnarray}&&{{a}_{n}}=\left\{\begin{array}{*{35}{l}} 1,\,\,L\left({{M}_{1}}+{{M}_{2}}\right)\leqslant n<L\left({{M}_{1}}+{{M}_{2}}\right)+{{M}_{1}} \\ 0,\,\,\text{else} \end{array},\,L\in N,\,M={{M}_{1}}+{{M}_{2}},\,\,\left\{{{M}_{1}},{{M}_{2}},M\in {{Z}^{+}}\right\}\right. \end{eqnarray} \tag{ 13 }$$

As shown in figure 4, ${{M}_{1}}$ and ${{M}_{2}}$ are the length of the testing TDLE sequence, corresponding to the numbers of ${{a}_{n}}=1$ (power ON) or ${{a}_{n}}=0$ (power OFF), respectively. Therefore the different ${{M}_{1}}$ and ${{M}_{2}}$ can reflect different dynamic power change characteristics with time. Thus we propose three modes of OOK TD power, i.e. transient, short and long power mode, as shown in table 1.

Table 1. Three modes of OOK TD power.

Mode	an  =  1:M1	an  =  0:M2	Current time duration	Dynamic power characteristics
Transient time	1–5	1–80	20 ms–100 ms	Fast impact power
Short time	5–50	5–300	100 ms–1000 ms	Mid-speed impact or fluctuating power
Long time	50–500	50–500	1 s–10 s	Low speed fluctuating power

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Based on the model of OOK TDLE sequence and three kinds of modes of TD power, we develop an OOK TDLE sequence generation equipment which can control TD power to change periodically, test dynamic error and compare testing result for smart meters.

A novel measure algorithm for dynamic error testing is proposed as follows:

We set ${{E}_{I}}(n)={{E}_{q}}\centerdot {{a}_{n}}$($n=0,\ldots,N$), $N=L\centerdot M$, where ${{E}_{I}}(n)$ is the dynamic energy sequence defined in equation (13), M is OOK modulating period, N is the length of the sequence and the positive integer L is the number of OOK period. Because the standard meter in figure 5 operates in the steady power state, during the N energy sequence, ${{a}_{n}}=1$ ($n=0,\ldots,N$) and the electric energy that has been measured by the standard meter is:

$$\begin{eqnarray}&&{{E}_{SO}}(N)=\underset{n=0}{\overset{N-1}{\sum}}\,{{a}_{n}}\centerdot {{E}_{q}}=N{{E}_{q}}=LM{{E}_{q}}\end{eqnarray} \tag{ 14 }$$

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On the other hand, the tested meter operates in the dynamic power state. During the N energy sequence, the numbers ${{a}_{n}}=1$ and ${{a}_{n}}=0$ are $L{{M}_{1}}$ and $L{{M}_{2}}$ respectively, so the theoretical dynamic energy sent to tested meter is:

$$\begin{eqnarray}&&{{E}_{XO}}(N)=\underset{n=0}{\overset{N-1}{\sum}}\,{{E}_{I}}(n)=\underset{n=0}{\overset{N-1}{\sum}}\,{{E}_{q}}\centerdot {{a}_{n}}=L{{M}_{1}}{{E}_{q}}\end{eqnarray} \tag{ 15 }$$

From (14) and (15), we obtain equation (16)

$$\begin{eqnarray}&&{{E}_{XO}}(N)=\frac{{{M}_{1}}}{M}{{E}_{SO}}(N)\end{eqnarray} \tag{ 16 }$$

It is easy to see the theoretical dynamic energy ${{E}_{XO}}(N)$ sent to tested meter can be traceable to steady electric energy ${{E}_{SO}}(N)$ measured by standard energy meter. Let the energy measured by tested meter is ${{E}_{X}}$, the relative dynamic error $\varepsilon$ of tested meter can be calculated as follows:

$$\begin{eqnarray}&&\varepsilon =\left({{E}_{X}}-{{E}_{XO}}(N)\right)\,/{{E}_{XO}}(N)\times 100\%\end{eqnarray} \tag{ 17 }$$

Relative dynamic error $\varepsilon$ is calculated based on the equations (14)–(17) and measured in our following testing system by comparing the electrical energy between tested meters and standard energy meter. The number L of OOK period is not necessarily an integer that causes a theoretical additional error which is not more than one OOK period number. Therefore the additional error caused by the measure algorithm (equations (14)–(17)) is less than $\gamma$:

$$\begin{eqnarray}&&\gamma =\frac{1}{L}\times 100\%\end{eqnarray} \tag{ 18 }$$

When OOK period L is not less than 300, the additional error is not more than 0.33%. Meanwhile, the OOK TDLE sequence is periodic, and this feature is suitable to repeat dynamic error testing and compare the testing result.

The block diagram of the testing system for dynamic error of the smart meter is shown in figure 5. It consists of standard energy meter (Radian Research RD33 Three-phase Analyzing Standard) which is used to get electric energy in the steady power state, standard power source (Fluke 6100A Electrical Power Standard), OOK TDLE sequence generation equipment (TDLE equipment) and a computer (Lenovo ThinkPad), two smart meter which are tested simultaneously.

To test the dynamic error of smart meters, first, on the computer set the programmable power source to generate three-phase steady AC voltage and current. Second, the three-phase steady voltage is fed to both the standard meter and OOK TDLE equipment which has been applied for patent in china, and meanwhile set the TDLE equipment operated at one TD power mode. Then the TDLE equipment generates three-phase OOK TD current which output to the two tested meters. Finally, the OOK TDLE equipment measures the output energy pulses from both the standard meters and the two tested meters, and calculates the two tested meters' dynamic errors which are shown at the LCD displayer of TDLE equipment. The dynamic error measure algorithm used in the TDLE equipment is given in section 3 of this paper.

Based on the testing system in figure 5, five kinds of meters are selected from different manufacturers for dynamic error testing. Meter A is a three-phase four-wire gateway meter manufactured by a European company; meter B is a three-phase four-wire smart meter from a Chinese company; meter C is a three-phase three-wire smart meter from a German company; meter D is a three-phase four-wire standard meter from an American company and meter E is a three-phase four-wire electronic meter from another Chinese company.

In the testing, we took ${{U}_{\text{rms}}}={{U}_{N}}\left(\alpha \text{=}1.0\right)$, ${{I}_{\text{rms}}}={{I}_{N}}\left(\beta =1.0\right)$, ${{\gamma}_{n}}=\cos {{\varphi}_{n}}=1.0$, and $L\geqslant 300$. All the meters were tested in three dynamic modes: transient, short and long mode, totally in six kinds of ON–OFF ratios (M₁:M₂). Table 2 shows the maximum, mean and minimum errors in five repeated testing.

Note that the dynamic errors are closely related to different dynamic power modes. For example, the dynamic errors of meter A in transient mode are almost 4 times higher than that in long mode. The dynamic error characteristics of meters from different manufacturers show diversity. The errors exhibited by these meters range from  −37.7% to 26.68%. In particular, some meters couldn't output pulse in transient mode, therefore, they are not suitable to measure the dynamic energy in transient mode.

Based on the the ISO GUM (guide to the expression of uncertainty in measurement), the uncertainty budget of the measurement results for smart meters is shown in table 3. ${{u}_{A1}}$ was evaluated by the statistical analysis of the repeated observations. The generated TDLE sequence contributes to the uncertainty of ${{u}_{A2}}$ which was evaluated by NIM (National Institute of Metrology, China). The standard uncertainty ${{u}_{B1}}$ caused by standard energy meter is less than 10⁻⁴. According to equation (18), the standard uncertainty ${{u}_{B2}}$ caused by the measure algorithm is 0.19% [24, 25].

Table 3. Uncertainty budget.

Source of uncertainty	Symbol	Type	Uncertainty contributions (%)
Repeatability of measurement	${{u}_{A1}}$	A	0.0028
TDLE equipment	${{u}_{A2}}$	A	0.0115
Standard energy meter (RD33)	${{u}_{B1}}$	B	0.0058
Measure algorithm	${{u}_{B2}}$	B	0.19
Combined uncertainty			0.192
Expanded uncertainty (coverage factor k  = 2)			0.38

In the paper the dynamic error testing problem for smart meter has been formulated, both the dynamic error testing method and testing signal were explored. First, an OOK TDLE model was established, through which two types of TDLE sequences and three modes of OOK testing dynamic power were proposed. Second, a measure algorithm to test dynamic error was derived, which provides a novel method for solving the problem of the dynamic electric energy measurement traceability to the electric energy measured by standard energy meter working in steady state. Third, based on the developed OOK TDLE sequence generation equipment and the method, a dynamic error testing system was constructed, and the dynamic errors for five kinds of smart meters were measured. Our results show that the dynamic error of smart meter is closely related to dynamic power mode and dynamic error characters vary from different kinds of smart meters. Finally, the measurement uncertainty was evaluated to be 0.38%. Based on the testing system and method discussed in the paper a general method was proposed for determining the dynamic characteristics of smart meters.

This work is supported by the National Natural Science Foundation of China (No. NSFC-51577006) and State Grid Jibei Electric Power Company Limited Research Program (No.52018K14001W).