Effect of harmonic distortion on electric energy meters of different metrological principles

Abstract

This paper deals with the errors of electric energy metering devices as a result of distortions in the shape of the curves of voltage and current load. It is shown and proved that the errors in energy measurements depend on the design and the algorithms used in electricity meters. There are three main types of metering devises having different principles: inductive (electro-mechanical), electronic static, and digital electronic (microprocessor). Each of these types has its measuring features. Some devices take into account all the harmonic distortions and the constant component which occur in the network while others measure the power and energy values of the fundamental harmonic only. Such traits lead to the discrepancies in the readings of commercial electric energy meters of different types. Hence, the violations in the measurement system unity occur, and a significant error can be observed in the balance of transmitted/consumed electric energy.

Keywords

current / distortion / electric energy meter / harmonics / power quality

Introduction

Three main types of electricity meters with different operating principles are used in the European Union (EU), Commonwealth of Independent States (CIS), and Ukraine for electric energy accounting: induction (electro-mechanical), electronic static, digital electronic (microprocessor).

Currently, there has been a stable trend toward the mass implementation of the latter two types. The latest achievements of microelectronics and digital signal processing methods are used in such metering devices. They provide high precision measurements in accordance with international (IEC) and government (DSTU and GOST) standards and perform a number of additional functions.

Electric meters, used currently, are designed and tested with the determination of their metrological characteristics. Thus, the mains voltage is believed to be of a sinusoidal (or close to it) waveform. On this basis, operation algorithms of metering equipment and their embodiment have been designed [¹]. However, in actual electric networks, the quality of electric power, including voltage and current, may significantly vary from the requirements of GOST 131901-97. In practice, those differences lead to the fact that the meters of the same nominal power (operable and verified) can show significantly different results of energy accounting consumed by the same load. In particular, that can be observed when comparing the readings of induction and static meters (case 1); induction and digital meters (case 2); two static meters(case 3); and two digital meters(case 4).

The difference in meter readings may exceed the sum of admissible error limits.

The objective of this paper consists in scientific and technical substantiation of the above-mentioned phenomena which occur during the operation of energy accounting meters based on a number of state-of-the-art research [¹,²], studying technical documentation and mathematical analysis of electricity meter algorithms.

Materials and methods

Induction electricity metering devices

The induction meter measuring mechanism is based on the creation of alternating magnetic voltage Ф_U and current Ф_I fluxes with a 90° phase-shift angle between them and directed perpendicular to a disk plane [¹,³].

The torque of the induction meter is defined by

(1)M_t=K{Ф}_U{Ф}_I\sin (\mathrm{90}°+\varphi ),

where K is the constant coefficient determined by the meter design, and ϕ is the phase angle between voltage and current.

Owing to iron saturation and hysteresis, the magnetic fluxes which are generated by voltage (Ф_U) and current (Ф_I) windings and cross the induction meter disk can be strongly distorted even at the sinusoidal current. In the presence of the higher harmonics in magnetic fluxes, the torque may significantly differ from the torque developed at sinusoidal magnetic fluxes. In accordance with the results of theoretical and experimental studies [⁴], torque-current curves of a complex shape lie between the sinusoidal curves at 50 Hz and 150 Hz. That is, the main influence on a disk torque is exercised by the 3rd harmonic. The influence of the other harmonics is either absent or negligibly small [⁴].

To obtain an induction meter almost not responding to higher harmonics or weakly dependent on them, manufacturers produce metering devices with such a constant K, where torques are equal for 50-period and 150-period current [⁴].

It follows that induction meter readings, with some approximation, are proportional to the active power of the fundamental harmonics of current and voltage load:

(2)W_P=U_{\mathrm{1}}·I_{\mathrm{1}}·\cos {\varphi }_{\mathrm{1}}·t=P_{\mathrm{1}}·t,

where U₁ and I₁ are effective values of the fundamental harmonic of the voltage and current, respectively, cosϕ₁ is the fundamental harmonic power factor, Р₁ is active power of the fundamental harmonic, and t is running time.

The difference between active and reactive energy meters is in the internal wiring circuit. Due to the connections of coils meant for 380 V, an additional 90° phase shift between magnetic fluxes is made. Implementation of single-phase reactive power meters is also performed by using a 90° shift circuit.

It follows that the readings of such an induction-type reactive power meter are proportional to the reactive power of the fundamental harmonics of current and voltage load:

(3)W_Q=U_{\mathrm{1}}·I_{\mathrm{1}}·\sin {\varphi }_{\mathrm{1}}·t=Q_{\mathrm{1}}·t,

where Q₁ is the reactive power of the fundamental harmonic.

Another point of view is proposed in the proceedings [³,⁵]. Researchers came to the conclusion that the ratio of the counter K (see Eq. (1)) is not constant and is a function of current, voltage, and angular frequency: K=f(U,I,\omega ).

If voltage and current are constant, K depends on the angular frequency only. Therefore, the presence of harmonics has an influence on the torque, which affects the readings of the meter.

According to Refs. [³,⁵], the error of an induction active energy meter can be determined by

(4)E\mathrm{\%}=\mathrm{100}\sum _{n>\mathrm{1}}\left(K\left(n\right)-\mathrm{1}\right)·E\left(n\right),

where K(n) is the function of the ‘constant’ of the meter from the angular frequency:

(5)K\left(n\right)={\displaystyle \frac{K_n}{K_{\mathrm{rat}}}},

where K_n is the value of K under the action of nth order harmonic, and K_rat is the value of K at rated parameters of voltage and frequency.

The empirical equation to determine it can be written as

K\left(n\right)=n·{\displaystyle \frac{\mathrm{1}+q_d^{\mathrm{2}}}{\mathrm{1}+n^{\mathrm{2}}{q}_d^{\mathrm{2}}}}·{\left({\displaystyle \frac{\mathrm{1}+q_v^{\mathrm{2}}}{\mathrm{1}+n^{\mathrm{2}}{q}_v^{\mathrm{2}}}}\right)}^{\mathrm{1}/\mathrm{2}}

(6)·{\displaystyle \frac{\mathrm{1}+a_n^{\mathrm{2}}{e}_{v\mathrm{0}}+b_n^{\mathrm{2}}{e}_{i\mathrm{0}}}{\mathrm{1}+a^{\mathrm{2}}{e}_{v\mathrm{0}}·{\displaystyle \frac{(\mathrm{1}+q_d^{\mathrm{2}})(\mathrm{1}+q_v^{\mathrm{2}})}{(\mathrm{1}+n^{\mathrm{2}}{q}_d^{\mathrm{2}})(\mathrm{1}+n^{\mathrm{2}}{q}_v^{\mathrm{2}})}}+b^{\mathrm{2}}{e}_{i\mathrm{0}}·{\displaystyle \frac{\mathrm{1}+q_d^{\mathrm{2}}}{\mathrm{1}+n^{\mathrm{2}}{q}_d^{\mathrm{2}}}}}},

where n is the number of a harmonic component, q_d, q_v, e_v0, and e_i0 are categorized as ‘meter parameters’ because they are associated with the internal structure of the metering device, q_d is the quality factor of the disk of the induction power meter, q_v is the quality factor of the voltage coil, e_v0 is the voltage braking torque ratio, and e_i0 is the current braking torque ratio. These parameters can be obtained from tests on the meter. The testing procedure must be provided while the meter is operating. The testing principles and proceedings for each parameter are listed in Ref. [³]. a_n and b_n are the nth voltage harmonic ratio and current harmonic ratio, respectively:

(7)a_n={\displaystyle \frac{U_n}{U_{\mathrm{rat}}}}={\displaystyle \frac{U_n}{U_{\mathrm{1}}}}·{\displaystyle \frac{U_{\mathrm{1}}}{U_{\mathrm{rat}}}}={\alpha }_n·a_{\mathrm{1}},

(8)b_n={\displaystyle \frac{I_n}{I_{\mathrm{rat}}}}={\displaystyle \frac{I_n}{I_{\mathrm{1}}}}·{\displaystyle \frac{I_{\mathrm{1}}}{I_{\mathrm{rat}}}}={\beta }_n·b_{\mathrm{1}}.

In Eq. (7), U_n, U₁, and U_rat are the nth harmonics, fundamental, and rated values of voltage, respectively. In Eq. (8), I_n, I₁, and I_rat are the nth harmonics, fundamental, and rated values of current, respectively. E(n) is the nth harmonic error factor and could be found by the empirical equation [⁶].

(9)E\left(n\right)={\displaystyle \frac{a_n·b_n·\cos {\varphi }_n}{{\displaystyle \sum _{m\ge \mathrm{1}}{a}_m·b_m·\cos {\varphi }_m}}}={\displaystyle \frac{{\alpha }_n·{\beta }_n·\cos {\varphi }_n}{\cos {\varphi }_{\mathrm{1}}+{\displaystyle \sum _{m>\mathrm{1}}{\alpha }_m·{\beta }_m·\cos {\varphi }_m}}}.

Given the above mentioned, Eq. (2) can be modificated to the form

(10)W_P=U_{\mathrm{1}}·I_{\mathrm{1}}·\cos {\varphi }_{\mathrm{1}}·t·\left(\mathrm{1}+{\displaystyle \frac{E\mathrm{\%}}{\mathrm{100}}}\right)=P_{\mathrm{1}}·t·\left(\mathrm{1}+{\displaystyle \frac{E\mathrm{\%}}{\mathrm{100}}}\right).

In this form of entry, the error of induction active energy meter (according to Refs. [³,⁵]) is taken into account.

Given that inductive meters of active and reactive power are similar in construction and operation, Eq. (3) with some approximation might be written as

(11)W_Q=U_{\mathrm{1}}·I_{\mathrm{1}}·\sin {\varphi }_{\mathrm{1}}·t·\left(\mathrm{1}+{\displaystyle \frac{E\mathrm{\%}}{\mathrm{100}}}\right)=Q_{\mathrm{1}}·t·\left(\mathrm{1}+{\displaystyle \frac{E\mathrm{\%}}{\mathrm{100}}}\right).

From Eqs. (10) and (11), it follows that the presence of higher harmonics has an effect on the readings of active and reactive power induction meters, and the presence of a constant component of current and/or voltage does not affect their readings. Meanwhile, according to Ref. [⁵], in most cases the error has a negative value.

Static electronic electricity metering devices

Electronic static and digital meters have an input voltage circuit in the form of a precision resistive sensor (voltage divider) with a high-impedance and low-impedance input. Electrical isolation on voltage input is absent. Therefore, a constant component of load voltage enters the meter circuit. Current input circuit is implemented in different ways in those meters [⁷] with a matching current transformer (TI), or with a matching current shunt (R_sh) in the main conductor circuit.

Thus, the meter circuit receives a complex shape voltage:

(12)u=U_{\mathrm{0}}+U_{\mathrm{1}m}·\sin (\omega t+{\varphi }_{\mathrm{1}})+U_{\mathrm{2}m}·\sin (\mathrm{2}\omega t+{\varphi }_{\mathrm{3}})+U_{\mathrm{3}m}·\sin (\mathrm{3}\omega t+{\varphi }_{\mathrm{5}})+\mathrm{...}

The effective (active) voltage value:

(13)U=\sqrt{U_{\mathrm{0}}^{\mathrm{2}}+U_{\mathrm{1}}^{\mathrm{2}}+U_{\mathrm{2}}^{\mathrm{2}}+U_{\mathrm{3}}^{\mathrm{2}}+\mathrm{...}+U_n^{\mathrm{2}}}=\sqrt{{\displaystyle \sum _{n=\mathrm{0}}^{\mathrm{40}}{U}_n^{\mathrm{2}}}},

where n is the number of a harmonic component,

U_n={\displaystyle \frac{U_{mn}}{\sqrt{\mathrm{2}}}}

is the nth harmonic effective voltage value, U_mn is the nth harmonic amplitude value, and U₀ is the voltage constant component value.

The maximum harmonic number is assumed to be 40 by analogy with Ref. [⁸].

The circuit of the meter with a shunt input receives current of complex shape:

(14)i=i_{\mathrm{0}}+i_{\mathrm{1}m}·\sin (\omega t+{\varphi }_{\mathrm{1}})+i_{\mathrm{2}m}·\sin (\mathrm{2}\omega t+{\varphi }_{\mathrm{2}})+i_{\mathrm{3}m}·\sin (\mathrm{3}\omega t+{\varphi }_{\mathrm{3}})+\mathrm{...}+i_{nm}·\sin (n\omega t+{\varphi }_n).

The effective current value

(15)I=\sqrt{{\displaystyle \frac{\mathrm{1}}{T}}{\displaystyle \underset{\mathrm{0}}{\overset{T}{\int }}{i}^{\mathrm{2}}\mathrm{d}t}}=\sqrt{I_{\mathrm{0}}^{\mathrm{2}}+I_{\mathrm{1}}^{\mathrm{2}}+I_{\mathrm{2}}^{\mathrm{2}}+\mathrm{...}+I_n^{\mathrm{2}}}=\sqrt{{\displaystyle \sum _{n=\mathrm{0}}^{\mathrm{40}}{I}_n^{\mathrm{2}}}}.

The effective current value excluding a constant component enters the meter circuit with a transformer input:

(16)I=\sqrt{{\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{40}}{I}_n^{\mathrm{2}}}}.

In reality, the constant component in the Fourier series will be absent only if the mean ordinate of the function equals zero over the whole period.

The active power of periodic alternating current of arbitrary waveform coming to the shunt meter input is defined as the average power for the whole period:

(17)P={\displaystyle \frac{\mathrm{1}}{T}}{\displaystyle \underset{\mathrm{0}}{\overset{T}{\int }}ui\mathrm{d}t}.

If instead of u and i, their expressions are substituted using trigonometric series (Eqs. (13) and (14)), then

P={\displaystyle \frac{\mathrm{1}}{T}}{\displaystyle \underset{\mathrm{0}}{\overset{T}{\int }}[U_{\mathrm{0}}+U_{\mathrm{1}m}·\sin (\omega t+{\varphi }_{\mathrm{1}})+U_{\mathrm{2}m}·\sin (\mathrm{2}\omega t+{\varphi }_{\mathrm{2}})}

+U_{\mathrm{3}m}·\sin (\mathrm{3}\omega t+{\varphi }_{\mathrm{3}})+\mathrm{...}+U_{nm}·\sin (n\omega t+{\varphi }_n)]

\times [I_{\mathrm{0}}+I_{\mathrm{1}m}·\sin (\omega t+{\varphi }_{\mathrm{1}})+I_{\mathrm{2}m}·\sin (\mathrm{2}\omega t+{\varphi }_{\mathrm{2}})

(18)+I_{\mathrm{3}m}·\sin (\mathrm{3}\omega t+{\varphi }_{\mathrm{3}})\mathrm{+}\mathrm{...}+I_{nm}·\sin (n\omega t+{\varphi }_n)]\mathrm{d}t.

The result is

P=U_{\mathrm{0}}·I_{\mathrm{0}}+U_{\mathrm{1}}·I_{\mathrm{1}}·\cos {\varphi }_{\mathrm{1}}+U_{\mathrm{2}}·I_{\mathrm{2}}·\cos {\varphi }_{\mathrm{2}}+U_{\mathrm{3}}·I_{\mathrm{3}}·\cos {\varphi }_{\mathrm{3}}

(19)+\mathrm{...}=\Sigma U_n·I_n·\cos {\varphi }_n={\displaystyle \sum _{n=\mathrm{0}}^{\mathrm{40}}{P}_n}.

Similarly, for reactive power:

(20)Q=U_{\mathrm{0}}·I_{\mathrm{0}}+U_{\mathrm{1}}·I_{\mathrm{1}}·\sin {\varphi }_{\mathrm{1}}+U_{\mathrm{2}}·I_{\mathrm{2}}·\sin {\varphi }_{\mathrm{2}}+U_{\mathrm{3}}·I_{\mathrm{3}}·\sin {\varphi }_{\mathrm{3}}+\mathrm{...}=\Sigma U_n·I_n·\sin {\varphi }_n={\displaystyle \sum _{n=\mathrm{0}}^{\mathrm{40}}{Q}_n}.

For meters with transformer input:

P=U_{\mathrm{1}}·I_{\mathrm{1}}·\cos {\varphi }_{\mathrm{1}}+U_{\mathrm{2}}·I_{\mathrm{2}}·\cos {\varphi }_{\mathrm{2}}+U_{\mathrm{3}}·I_{\mathrm{3}}·\cos {\varphi }_{\mathrm{3}}+\mathrm{...}

(21)=\Sigma U_n·I_n·\cos {\varphi }_n={\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{40}}{P}_n},

Q=U_{\mathrm{1}}·I_{\mathrm{1}}·\sin {\varphi }_{\mathrm{1}}+U_{\mathrm{2}}·I_{\mathrm{2}}·\sin {\varphi }_{\mathrm{2}}+U_{\mathrm{3}}·I_{\mathrm{3}}·\sin {\varphi }_{\mathrm{3}}+\mathrm{...}

(22)=\Sigma U_n·I_n·\sin {\varphi }_n={\displaystyle \sum _{n=\mathrm{1}}^{\mathrm{40}}{Q}_n}.

In accordance with Eqs. (19) and (21), in order to determine the value of P in measuring chips (MC) of static meters, there occurs the multiplication of functions represented in the form of electric pick-off signals. In the meters with transformer input (TI), the low-pass filtering that suppresses a variable component is performed after that. The value of the consumed energy in Watt-hours is obtained by means of integration. The electromechanical indicator with a stepper motor externally connected to the chip can serve as an integrator. The motor is fed by pulses from the MC output, the repetition frequency of which is proportional to the active power P.

For measuring the reactive power by using Eqs. (20) and (22), an additional 90° phase shift between voltage and current functions is introduced. In other respects, the reactive power measurement is similar to the active power measurement: the voltage and current functions (with the specified 90° shift) are multiplied, and the function of instantaneous reactive power (instantaneous values) is obtained. Next, filtration is performed, and the value proportional to the mean value of reactive power Q is received.

From Eqs. (19)–(22) it follows that the readings of electronic static meters with different circuit schemes may be different depending on the presence or absence of a constant voltage or current component. This explains the differences for electronic static active energy meters described earlier in this paper (case 3).

From Eqs. (19) and (21) it follows that the readings of electronic static active energy meter in all cases will exceed the readings of inductive electric energy meter. The only exception is when network voltage and current are of a sinusoidal shape. In addition, a constant component must be also unavailable for shunt meters.

Equations (20) and (22) lead to the similar conclusion for electronic static and induction-type reactive energy meters. That explains the statement (case) 2 mentioned at the beginning of this paper.

Note that at finding the apparent power by Eq. (23).

(23)S=\sqrt{P^{\mathrm{2}}+Q^{\mathrm{2}}}.

Using Eqs. (19) and (20) or (21) and (22), the result will differ from the real power coming to the system input.

That is,

(24)S=UI=\sqrt{U^{\mathrm{2}}{I}^{\mathrm{2}}{\cos }^{\mathrm{2}}\varphi +U^{\mathrm{2}}{I}^{\mathrm{2}}{\sin }^{\mathrm{2}}\varphi }>\sqrt{P^{\mathrm{2}}+Q^{\mathrm{2}}}=\sqrt{{(\Sigma U_n·I_n·\cos {\varphi }_n)}^{\mathrm{2}}+{(\Sigma U_n·I_n·\sin {\varphi }_n)}^{\mathrm{2}}}.

This difference can be explained by the distortion power arising in networks at non-sinusoidal voltage and current waveforms [⁶,⁹,¹⁰].

The square of distortion power is defined as the squared voltage multiplied by the sum of current harmonics excluding the fundamental wave.

(25)A_{cn}=UI=\sqrt{\Sigma [U_k^{\mathrm{2}}{I}_n^{\mathrm{2}}+U_n^{\mathrm{2}}{I}_k^{\mathrm{2}}-\mathrm{2}{U}_k{I}_k{U}_n{I}_n\cos ({\varphi }_k-{\varphi }_n\left)\right]},

where the indices k and n take values 1, 2, 3,... independently from each other, but k does not equal n. Then, the total power can be represented as

(26)\begin{array}{l}S=\sqrt{P^{\mathrm{2}}+Q^{\mathrm{2}}+A_{cn}^{\mathrm{2}}}\hfill \\ =\sqrt{{(\Sigma U_n·I_n·\cos {\varphi }_n)}^{\mathrm{2}}+{(\Sigma U_n·I_n·\sin {\varphi }_n)}^{\mathrm{2}}+\Sigma [U_k^{\mathrm{2}}{I}_n^{\mathrm{2}}+U_n^{\mathrm{2}}{I}_k^{\mathrm{2}}-\mathrm{2}{U}_k{I}_k{U}_n{I}_n\cos ({\varphi }_k-{\varphi }_n\left)\right]}.\hfill \end{array}

Distortion power is not subject to measurement and can be determined only by means of calculations [⁹]. It is a component of apparent power that exists only at nonlinear load [⁹]. The distortion power equals zero only when a shift angle between voltage and current is the same for all the harmonics, as well as when the ratio between effective values of current and voltage is of the same value for all the harmonics [¹¹]. It is possible only with the purely active load nature.

The algorithms of electronic static-type meters take into account the distortion power [⁶,¹²]. They often use Eq. (27) for computing.

(27)Q=\sqrt{S^{\mathrm{2}}-P^{\mathrm{2}}}.

Thus, even when no one harmonic has reactive power intake, the meter may indicate some reactive power consumption. This once again confirms the conclusion made for the 1st phenomenon (case 1) for reactive electric energy meters.

Electronic digital electricity metering devices

The significant difference between electronic digital and electronic static electric energy metering devices is that digital ones have equal circuits for both active and reactive energy meters. In the simplest case, their operation is implemented with the help of digital signal processing (DSP) which performs all the necessary transformations by measuring the instantaneous current and voltage values at discrete time intervals. The signals proportional to the current and voltage values received from the respective sensors are delivered to DSP inputs [¹].

The electronic circuit is a set of several fast-responding analog-digital converters of instantaneous input currents and voltages.

Single-phase devices have one pair of converters, three-phase ones—three pairs of converters (power consumption is measured separately on each phase). During the period of power reference frequency, an even number (2n) of pairs of instantaneous values of current i_n and voltage u_n is measured (n = 0, 1, 2,..., N).

The discretized values of voltage and current (i_n and u_n) are processed in the microprocessor which is a part of a meter for obtaining effective values of current and voltage, active and reactive power, etc. It is obvious that the measurement accuracy increases with the sampling frequency. That, in turn, results in software meshing because processing is performed in real time.

Metrological properties and errors of electronic energy meters depend on computing methods and features of the programs implemented in their microprocessors. In general, all the calculation algorithms can be divided into two groups, those based on the use of the Fourier transforms [¹³,¹⁴], and those based on the electrical methods of power determination [¹¹,¹²,¹⁵,¹⁶].

Let us consider the algorithms based on the Fourier transforms.

It is known that any periodic function f(x) that satisfies the Dirichlet conditions, i.e., if over a period of time a function has a finite set of discontinuities of the first kind (ordinary discontinuities) and a finite set of maxima and minima, can be represented in the form of a harmonic series:

(28)f\left(x\right)=A_{\mathrm{0}}+A_{\mathrm{1}}\sin x+B_{\mathrm{1}}\cos x+A_{\mathrm{2}}\sin \mathrm{2}x+B_{\mathrm{2}}\cos \mathrm{2}x+\mathrm{...}

or

(29)f\left(x\right)=A_{\mathrm{0}}+{\displaystyle \sum _{k=\mathrm{1}}^{\infty }}(A_k\sin kx+B_k\cos kx),

where the coefficients of Fourier series are determined by

(30)A_{\mathrm{0}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}{\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{2}\mathrm{\pi }}{\int }}f\left(x\right)\mathrm{d}x\approx {\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{k=\mathrm{0}}^{\mathrm{2}n-\mathrm{1}}{f}_k\left(x\right)=}{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{k=\mathrm{1}}^n{f}_k\left(x\right)},}

(31)A_k={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}{\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{2}\mathrm{\pi }}{\int }}f\left(x\right)\sin kx\mathrm{d}x={\displaystyle \frac{\mathrm{2}}{n}}{\displaystyle \sum _{k=\mathrm{1}}^n{f}_k\left(x\right)\sin \left(km{\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{n}}\right)},}

(32)B_k={\displaystyle \frac{\mathrm{1}}{\mathrm{2}\mathrm{\pi }}}{\displaystyle \underset{\mathrm{0}}{\overset{\mathrm{2}\mathrm{\pi }}{\int }}f\left(x\right)\cos kx\mathrm{d}x\approx {\displaystyle \frac{\mathrm{2}}{n}}{\displaystyle \sum _{k=\mathrm{1}}^n{f}_k\left(x\right)\cos \left(km{\displaystyle \frac{\mathrm{2}\mathrm{\pi }}{n}}\right),}}

where n is the number of counts of instantaneous values (voltage or current) over the period of fundamental frequency, f_k(х) determines the instantaneous voltage or current value for the numbered count k, and m is the number of the harmonic for which the count is carried out.

Then, it is possible to calculate any parameters of current, voltage and power in different versions:

(1) Only for main harmonic: a meter similar in its properties to an induction-type meter will be obtained;

(2) For several harmonics, excluding the constant component (if a transformer in the circuit is stipulated): a meter, similar in its properties to a static electronic meter with a transformer input (TI) will be obtained;

(3) For several harmonics, including a constant component (if a shunt in the current circuit is stipulated): a meter, similar in its properties to a static electronic meter with a shunt input (R_sh) will be obtained.

The above implies that digital meters may have the algorithms similar to induction and electronic static metering devices working according to the above-described algorithm. The algorithm embedded in the programmable chip of a particular digital meter is not disclosed by manufacturers. Manufacturers attribute this to the fact that the developed algorithm is the product of intellectual property, and it is a trade secret. This can explain phenomenon 2 described at the beginning of this paper. In addition, different digital meters of the same accuracy class can have different algorithms. This fact can easily explain phenomenon 4 (case 4).

Next, the algorithms based on the conventional formulas of electrical engineering are considered.

Active power value can be obtained by

(33)P={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{k=\mathrm{0}}^{n-\mathrm{1}}{P}_k},

where f_D is the sampling frequency, f_C is the network frequency, and n = f_D/f_C is the number of counts in one period of the measured signal.

Then Eq. (34) can be written.

(34)P={\displaystyle \sum _{k=\mathrm{0}}^{\mathrm{40}}{P}_k},

where

P_k=U_k·I_k·\cos {\varphi }_k.

The reactive power can be calculated by Eq. (27) or by the instantaneous values of current and voltage, similar to Eq. (34).

(35)Q_k={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \underset{\mathrm{0}}{\overset{t}{\int }}u\left(t\right)·i(t-{\displaystyle \frac{n}{\mathrm{4}}})\mathrm{d}t={\displaystyle \sum _{k=\mathrm{0}}^{\mathrm{40}}{Q}_k},}

where

Q_k=U_k·I_k·\sin {\varphi }_k.

When calculating the reactive power according to Eq. (27), the complex power is assumed as

(36)S=U·I.

From the above and from Eq. (25), it follows that calculations by Eqs. (21) and (34) or (22), (27) and (35) will give the same result only for purely sinusoidal currents and voltages without the constant component.

The presence of distortions in waveform or in a constant component leads to different results for the meters that employ different calculation algorithms. This once again confirms statement (case) 4 adduced at the beginning of this paper.

Estimated verification of the findings

Let us confirm the above conclusions through mathematical calculations. Suppose that there is an electric receiver with the miscellaneousnature of load (active-reactive with prevalence of reactive inductive). Let’s assume that the voltage and the current have a harmonic composition, which is specified in Table 1. The effective values of the voltage and the current at the harmonic composition given in Table 1 are equal to 212.2 V and 15.35 A, respectively.

The coefficients of voltage harmonic components K_Un are taken in accordance with the requirements in Ref. [⁸] for single-phase 380 V networks. With the selected values K_Un the nonsinusoidal voltage coefficient K_m amounts to 10.73% which does not exceed the admissible 12% established in Ref. [⁸].

The coefficients of current harmonic components K_In have been taken twice as high as that for the voltage. However, the level of nonsinusoidal current makes 21.45%. At the given mode of voltage it is quite possible, since the distortions of the sine current wave in the load are stronger than the distortions of the voltage sine wave [²,¹⁷].

Let us note that in Ref. [⁸] a current waveform distortion coefficient is not standardized.

It is also assumed that the constant voltage and the current components are contained. In Ref. [⁸], there are not any requirements for them, though their presence may significantly affect the result of meters’ work with some of the considered algorithms.

The dependencies of the instantaneous voltage and current values with the accepted harmonic composition on the time are shown in Fig. 1.

Table 2 lists the calculated values of load power indicators for voltage and current specified in Table 1. Calculations have been made in accordance with the above-described algorithms of various electricity metering devices.

Apparent power can be calculated in two ways. In Table 2, S′ is determined by

(37)S\text{'}=\sqrt{P^{\mathrm{2}}+Q^{\mathrm{2}}}.

Complex power S" is given by Eq. (31). As it is shown in Eq. (21), the results of calculation may mismatch in two cases.

The data in Table 2 visually demonstrate how the results of electric energy accounting can differ when using various modifications of modern meters even at relatively small distortions of voltage (within GOST 131901-97) and current waveform.

The largest divergence in active power indications is 5.99%. The largest divergence in reactive power indications is equal to 4.38%. The largest divergence in apparent power is 4.79%.

In operating electric networks where the harmonic composition can exceed the limits specified in Ref. [⁸], the discrepancies can be much larger. Such differences in commercial electric meter readings result in the violation of measuring system uniformity and introduce significant errors in the balance of transmitted/consumed electric energy.

Electric energy accounting mistakes, caused by the presence of higher harmonics, often arise in networks with valve converters (mostly thyristor) [⁶]. A good example is electricity accounting in power supply circuits with induction motors fed from PWM inverters in Ref. [¹⁸].

Pulse width modulation (PWM) is a technique in which a fixed input DC voltage is given to the inverter and a controlled AC output voltage is obtained by adjusting the on and off periods of the inverter components. The block diagram of a voltage source inverter fed induction motor drive is displayed in Fig. 2.

The system configuration of a boost AC/DC/AC rectifier is demonstrated in Fig. 3. In this draft R_1, R_2, R₃ and L₁, L₂, L₃ are impedances and inductances of the supply lines, respectively; T_1s, T_2s, T_3s and T_1i, T_2i, T_3i are IGBT-transistors; C is the filter, u₁, u_2, u₃ are voltages of the 3-phase source, i₁, i_2, i₃ are currents in each phase, i_c is the current in the filter’s brunch; and i_L is the current in the load brunch.

The load on the AC side is the induction motor.

To encounter the above discussed phenomena (meters show significantly different results of energy accounting consumed by the same load) the circuit should be analyzed under the assumptions that the input AC voltage (source) is a balanced three-phase supply [¹⁸], the power switches are non-ideal and generate higher harmonics, the PWM-inverters are nonlinear while other circuit elements can be linear and time invariant, and the electric energy meters of different types are installed in the point of balance accessory, before the consumer’s grid/before the load.

Conclusions

It is theoretically proved and practically confirmed that electricity meters of various types with the same rated power (operable and verified) can show significantly different results of power consumption for the same load. The difference of readings can exceed the amount of admissible errors of meters. This is explained by the following facts:

(1) The meters of induction type, due to their design features, almost ignore the power of higher harmonics of current and voltage, which are multiple of three (3rd, 6th, 9th, 12th,15th). But they are sensitive to other odd harmonics, particularly the 5th, 7th, 11th, 13th and 17th [⁶].

(2) The static type electronic meters with a matching current shunt (R_sh) take into account the active and reactive power of the higher harmonics and the active power caused by the presence of a constant component in current and voltage curves. Static meters with a matching transformer (TI) take into account of the higher harmonics power, but do not allow for the power caused by the presence of a constant component.

(3) The calculation algorithms of electronic meters can be based on the Fourier transforms or on electrical engineering methods of power determination. The algorithm, embodied in a programmable chip of a particular digital meter type, is considered as “know how,” and not revealed by manufacturers.

(4) In the manuscript it is demonstrated how the results of electric energy accounting can differ when using various modifications of modern meters even at relatively small distortions of voltage and current (within established standards) waveform. For the determined harmonic composition of load voltage and current, the largest divergence in active power indications is 5.99%, the largest divergence in reactive power indications is equal to 4.38%, and the largest divergence in apparent power is 4.79%.

These phenomena can be supervised in power supply circuits with induction motors fed from PWM inverters.

The differences in the readings of commercial electric energy metering devices lead to the violation of the uniformity of measurement systems and introduce significant errors in the balance of transmitted/consumed electric energy.

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