Analysis about effects of DC ripple on DC energy meter

Preface

The measurement and traceability of DC energy meter has no reference standard at present. One approach is to use a DC voltmeter and a DC ampermeter measuring the RMS and then multiply to obtain the DC power then compute energy. Another option would be to sample the value of voltage and current  at the same time, accumulate the instantaneous power electricity to obtain energy. If no ripple DC signal, that is,  no superposition any ac signal, both way are accurate in theory, so the traceability of DC energy can also be simplified as raising the level of the dc voltage and dc current to a separate source. But the reality is superposition of ac signal inside some of the dc signal is unavoidable, so whether the above two methods have error for the measurement of the dc watt-hour meter and what they are, let`s analyze the errors in the following, so as to provide reference for dc traceability and dc measurement.

I. Error analysis----multiply value of voltage and current to obtain energy

For dc power, there is no reactive power, so there should be only active power. But due to the dc superimposed ac signal, actually reactive power will occur. So when analyzing the error, we introduce many concepts in order to understand standard calculation formula of active power easier.

1-1

Formula 1-1: Multiply RMS to obtain active power

1-2

Actually, formula 1-2: apparent power
For dc signal superimposed ripple, any continuous periodic signal can be made of Fourier series. So dc current or voltage with ripple can be shown in formula 1-3, 1-4

1-3

1-4

Formula 1-5 and 1-6 : According to the definition of RMS

1-5

1-6

Insert formula 1-3 and 1-4 into 1-5 and 1-6 , we can obtain RMS

1-7

1-8

Therefore, RMS of dc ripple (non sinusoidal periodic voltage or current) is equal to the square of its constant component and sum of the square of every harmonic RMS and then square root.
Insert formula 1-3 and 1-4 into formula 1-1, we can get the following active power:

1-9

According to Orthogonal principle, instantaneous value`s product of sine wave is equal to zero in one period, after we accumulate the values, we can get:

1-10

1-11

We know that the reactive power in the sine circuit indicates the maximum of electromagnetic energy exchange.But in cases of dc ripple (non sinusoidal) , according to the calculation above, because it can be in any of the four quadrant, which can be positive or negative, so all may offset each other , even Q is zero. However, the reactive current component is not equal to zero, which means it dose`t stand for physical meaning, such as, largest energy exchange and so on. Q stands for the sum of reactive power caused by phase difference in every harmonic voltage and current.Multiply RMS of voltage and current to get active power, that is, insert formula 1-7 and 1-8 into formula 1-2, we get,

1-12

Actually  Pd = S; S is apparent powerObviously,
But, 
Let alone,
Therefore, for the case of dc energy with dc ripple, if we only multiply the RMS, it will cause more active energy and the result is inevitable. The excess energy can be calculated in the following way:

1-13

when, we can obtain maximum  , which is

1-14

Well, the truth iscan`t always appear. Because the mathematical expression of is not very intuitive, it is difficult to evaluate the largest error of  from the formula. However, based on formula 1-14, we can estimate the relative error,

1-15

If the convergence of the ripple is the h-th wave, the ratio of ripple and dc current and voltage is 
.
The power factor angle of h-th harmonic wave is ranging from 0° to 360°.
The range of is 0-100%.
The distribution diagram of simulation error through mat-lab is listed as follows:

Chart 1-1 the distribution diagram of energy error

when the ripple content of voltage and current is up to 100%, adopt RMS to calculate dc energy.

From the mat-lab simulation, it indicates that when harmonic power factor angle is near 180, and UhIh is close to 100%, error will be very big. because true value of P is close to zero, the denominator is very small,besides, there is no more accurate simulation, When the power factor Angle equals to 180° and Uh = 100%, its error is infinity.When the range is 0-10%, through mat-lab simulation error distribution diagram is as follows

Chart 1-2 the distribution diagram of energy error

when the ripple content of voltage and current is up to 10%, adopt RMS to calculate dc energy.
From chart 1-2, it shows that once the harmonic wave content decreases, the P value of  reduces slightly, the denominator is very large, the error fluctuate a little. To sum up, when the upper limit of harmonic wave content is 10%, the error is less than 2%.
When the range of is 0-1%, the error distribution error is listed as chart 1-3,

Chart 1-3 the distribution diagram of energy error

when the ripple content of voltage and current is up to 10%, adopt RMS to calculate dc energy.
As is shown above, once the harmonic wave content decreases，when the upper limit of harmonic wave content is 1%, the error is less than 0.02% to 1% of the installation type dc watt-hour meter as long as the voltage and current ripple is less than 1%, its error, the error of using the RMS of current and voltage to multiply , is negligible.

II. Error analysis- Synchronous sampling to measure DC energy

The hardware of dc energy meter can be simplified as chart 2-1. Generally, it is made up of voltage converter, current transformer, Low-pass filter, AD sampling and CPU.
Based on the approach of accumulating instantaneous power, this part mainly analyzes the effects of  the hardware filter and AD non synchronous sampling on dc energy. At the same time, suppose the cut-off frequency of LPF filter is less than sampling rate Fs/2 (to guarantee no superimposed  Frequency spectrum) .

Chart 2-1 design diagram---dc measuring the hardware

Chart 2-1 circuit function (neglect the effect of voltage and current change on error)

2-1

X(S): voltage or current input signal
: Simulate front-end low-pass filter transfer function(butter-worth stands for simulation second low-pass filter)

: AD sampling time-lapse transfer function
Y(S): AD final collected voltage or current data

The original input signal is P1

After LPF and sampling time-lapse, the signal power is P2
Thus, the error is,

2-2

The detailed derivation of error formula will be complemented in the subsequent development.
Make quantitive analysis to filter wave and synchronous time-lapse through mat-lab simulation.

Mat-lab simulation error analysis when using hi-speed AD

Adopting hi-speed chip in AD circuit, the sampling frequency is 100KHZ(similar to ADS 2374 regardless of the sampling principle)
The ripple frequency spectrum of input signal is focused on 10-50KHZ (chart 2-2)
The cut-off frequency of voltage channels using second low-pass filter is 10KHZ.
The cut-off frequency of current channels using second low-pass filter is 1KHZ-10KHZ.
Sampling time-lapse of voltage channel is 0
Sampling time-lapse of current channel is 0-1/fs
Error via mat-lab simulation

Chart 2-2  simulation wave

When the ripple content is 10%, the error distribution diagram 

Chart 2-3 error distribution diagram--100K sampling ripple content is 10%

From the chart above, when ripple is 10%, its error is ranging from -1.5% to -0.5%. because of the  amplitude frequency, the system error is inclined to - 0.5%, this can be corrected by the software, so when using hi-speed synchronous sampling, harmonic content 10%, the system error is about 1%

Error distribution diagram--ripple content is 1% (hi-speed AD)

Chart 2-4 100K sampling frequency when ripple content is 1%

From chart above, when ripple is 1%, the error is about -0.01%, which requires the error class installation type energy meter is 1%(can be negligible)

Mat-lab simulation error analysis--when using low-speed AD

Adopting low-speed chip in AD circuit, the sampling frequency is 100KHZ(similar to ADS 7793 regardless of the sampling principle)
The ripple frequency spectrum of input signal is focused on 10-50KHZ
The cut-off frequency of voltage channels using second low-pass filter is 20KHZ.
The cut-off frequency of current channels using second low-pass filter is 10KHZ-40KHZ.
Sampling time-lapse of voltage channel is 0
Sampling time-lapse of current channel is 0-1/fs
Error via mat-lab simulation

Error distribution diagram--when ripple content is 10% (low speed AD)

Chart 2-5 error distribution diagram-- 100K sampling frequency when ripple content is 1%

From the chart above, when the ripple is 10%, it has little influence on the LPF low-pass filter for the frequency signal is mainly concentrated on high frequency. And the influence of sampling time-lapse Td is similar to chart 2-3. Because when the sampling frequency is 100HZ, Td ranges from 0 to 100000uS. However, when it is 100KHZ, the range of Td is 0-10uS, therefore, compared with chart 2-3, it has smaller influence.

Error distribution diagram--when ripple content is 1% (low speed AD)

Chart 2-6 error distribution diagram-- 100K sampling frequency when ripple content is 1%

From chart above, when ripple is less than 1%, the influence of nonsynchronous and LPF is less than 0.01%, which can be negligible for installation type energy meter.