Abstract
PMI recorders use a variety of algorithms to compute RMS voltage and current, real, reactive, and apparent power, true and displacement power factor, and phase angle. The formulas for these algorithms are detailed here.
1. Introduction
A PMI recorder samples up to four pairs of voltages and currents. From these samples it computes RMS voltage and current; real, reactive, and apparent power; power factor and displacement power factor, phase angle, voltage and current THD, and harmonic magnitudes and phases. The Vip, 600P, Eagle, Eagle 120, and Guardian sample the raw waveforms at rate of 256 samples per powerline cycle (usually 60Hz). The iVS-2S samples at 128 samples per cycle. The Revolution and Vision sample voltage at a 1MHz rate (one million samples per second), and current at 250,000 samples per second. These high-speed data streams are downsampled to 256 samples per cycle for RMS and power calculations.
The complications of A/D quantization, scaling, finite precision math, gain and offset correction, hardware temperature drift compensation, harmonic magnitude and phase corrections, and synchronization with the powerline frequency are not discussed here. Thus, assume all measurements are in volts or amperes, with infinite precision, and perfectly synchronized such that 256 samples is exactly one powerline cycle (hereafter called a 60 Hz cycle, though the actual frequency may be from 46 to 70 Hz). The formulas given here are not necessarily those performed by the recorder, but are numerically equivalent expressions.
1.1 Notation and Sampled Data
Up to four channels of voltage and four channels of current are sampled. Let v1[n], v2[n], v3[n], v4[n] and i1[n], i2[n], i3[n], i4[n] represent the sampled voltages and currents for the four channels. In a single 60Hz cycle, the samples are indexed in the range 0 ≤ n ≤ 255. Where the channel number is not relevant, the subscript may be dropped. Where multiple cycles of data are needed, a superscript is added: vjm[n] is the nth voltage sample for the jth channel for the mth cycle, where 0 ≤ n ≤ 255, 1 ≤ j ≤ 4, and m > 0. For recorders which sample at 128 points per cycle, any summation below to 256 extends only to 128 samples.
2. Independent Channels/Single Phase
In this recording mode, each pair of voltage and current channels are used independently. Three phase wye and delta calculations are extensions to the formulas for the single phase case. For the Eagle 120, one pair of voltage and current channels are used for power calculations, and the second voltage channel (neutral to ground) is only used in RMS calculations.
2.1 RMS Voltage and Current
The RMS value is computed once per cycle for each channel of voltage and current. The voltage RMS value is computed by:
Similarly, the current RMS value is given by:
2.2 Real Power
Real power is computed once per cycle for each pair of voltage and current channels. The real power value is computed by:
NOTE: Real power is signed to indicate direction of power flow.
2.3 Apparent Power
Apparent power is computed once per cycle for each pair of voltage and current channels. The apparent power value is computed by:
2.4 Harmonics
An FFT of each voltage and current channel is computed every cycle. Since harmonics only to the 51st are required, the anti-aliased, sampled data is smoothed and downsampled by a factor of two before a 128-point FFT is performed, for recorders that sample at 256 points per cycle. The smoothing is done by averaging each pair of data points. The complex FFT result, including the smoothing and downsampling, is given by:
for k=0,…, 63. Here j represents √–1. Since the FFT is done on a single 60Hz cycle of data, the index k also represents the harmonic number. The 128 point FFT gives a decomposition into 64 harmonics of 60Hz. For specific channels and cycle numbers, the notation Vjm[k] and Ijm[k] denote the FFT value for jth channel, for the mth cycle number, for the kth harmonic. The real and imaginary parts of V[k] are denoted by Vx[k] and Vy[k], respectively. The real and imaginary parts for channel j are Vjx[k] and Vjy[k].
The harmonic magnitudes and phases are computed once per second, to provide some averaging and to reduce transient effects. The one-cycle FFT values are averaged over the M cycles which comprise each second, to form:
the kth harmonic magnitude is then given by:
and the raw kth harmonic phase angle is:
The arctan function is the four quadrant inverse tangent, with a range of -180 to +180 degrees. The current magnitudes and phase angles are computed in the same manner. The voltage harmonic phase angles are referred to the first voltage channel’s first harmonic phase angle. The current harmonic phase angles are then referred to their corresponding voltage 60Hz phase angles. This two-step algorithm proceeds as follows for the jth channel:
2.5 Phase Angle
The phase angle, θ, is the angular phase shift between the 60Hz voltage and current sinusoids. It is computed every cycle, and is simply:
where Iθ[1] and Vθ[1] are the phase angles for the 1st harmonic (60Hz). These phase angles are computed using (8) on the raw FFT outputs instead of the one second average, with k = 1.
2.6 Reactive Power
Reactive power is computed every cycle for each pair of voltage and current channels. The result is given by:
Each Vx[k]Iy[k] – Ix[k]Vy[k] term is the reactive power contributed by harmonic k. This can also be expressed for each harmonic as VRMS × IRMS × sin θ, where VRMS, IRMS, and θ are the values for RMS voltage and current at harmonic k, and the sine of the phase angle between the voltage and current at harmonic k. A similar expression exists for computing real power; this expression reduces to (3).
2.7 Power Factor
Power factor is computed once per cycle for each pair of voltage and current channels. The result is given by:
NOTE: The power factor is neither leading nor lagging if θ = 0 or θ = ±180. This expression is also known as true power factor, since it includes the effects of harmonics.
2.8 Displacement Power Factor
Displacement power factor is computed once per cycle for each pair of voltage and current channels. This quantity represents only the 60Hz contribution to the true power factor. The result is computed by:
NOTE: The displacement power factor is neither leading nor lagging if dPF = 1. If dPF = PF, then voltage and current harmonics must be present. However, even when dPF = PF, harmonics may still be present.
2.9 THD
Total harmonic distortion, computed every second for each channel of voltage and current, is given in percent by:
Since this THD definition is referred to the fundamental (as opposed to the RMS value), it may be over 100%.
THD is computed separately every cycle in the Revolution and Vision, for THD waveform capture triggering. These cycle THD values are not used to form longer THD averages. Due to the nonlinearity of the THD formula, averaging the cycle THD values over one second is not mathematically identical to computing THD over averaged harmonic magnitude values.
3. Three Phase Wye
In a three phase wye hookup, each pair of voltage and current channels are handled in the same manner as the single phase hookup. The first three pairs are also grouped together to form total power quantities.
3.1 Total Powers
Total real, reactive, and apparent power are computed and displayed but not recorded in wye mode. The three phase totals are the sum of the individual phases:
All these totals are computed every second from one second averages. The values are displayed on the front panel and then discarded.
3.2 Total Power Factors, Phase Angle
These total quantities are computed as weighted averages of the three phases, weighted by apparent power:
All these totals are computed every second from one second averages. The values are displayed on the front panel and then discarded.
4. Three Wire Delta
With a three wire delta circuit, individual phase powers and power factors cannot be computed without imposing assumptions such as a balanced load, balanced source, etc. Only total quantities can be computed in this mode. These values are computed and recorded as channel one data. As in the wye case, these values are computed once per cycle. The fourth channel, if present, is treated as an extra single phase channel with power calculations as detailed in Section 2. Real and reactive power are calculated using the two-wattmeter method, using voltage channels 1 and 2, and current channels 1 and 3. A Vip or 600P, with differential inputs, is connected as a delta, with each voltage channel connected from phase to phase. A Revolution, Vision or Eagle, with a common, is connected with each voltage input on a separate phase, and the common on B delta corner. In this case, delta voltages must be computed from the raw voltage input measurements:
where v̂1, v̂2, and v̂3 are the raw voltage samples as measured on the inputs, all referred to the same common. In the standard delta hookup, v̂2 will be zero, since the common (white) input will be connected to the same point as the second voltage input. Since a true delta circuit must satisfy v1[n] + v2[n] + v3[n] = 0, the conversion from v̂ to v is exact, and requires no assumptions about balanced loads, harmonics, etc. After this conversion, v̂1, v̂2, and v̂3 are used in all other formulas as if measured directly.
4.1 Real Power
Real power is computed using the two-wattmeter method. This requires two voltage and current channels to compute the three phase total. Voltage channels one and two are used with current channels one and three:
4.2 Reactive Power
Reactive power is computed using the two-wattmeter method. This requires two voltage and current channels to compute the three phase total. Voltage channels one and two are used with current channels one and three:
This is a two-wattmeter extension of the single-phase reactive power method (10).
4.3 Apparent Power
Apparent power is computed by the vector method.
The Revolution and Vision can be configured to use the arithmetic apparent power method. With this method, (16) is used to sum three individual VA values. To compute these values, an artificial neutral is assumed at the centroid of the vector triangle formed by the measured phase voltages. This neutral is then used to compute RMS line-neutral voltages:
where V1, V2, and V3 are RMS voltage values from (1). The line-neutral voltages are then used to compute individual phase apparent power:
From this, (16) is used to form the arithmetic sum.
4.4 Phase Angle
The phase angle, θ, is the angular phase shift between the 60Hz voltage and current sinusoids. Since the actual phase current cannot be measured in a three wire delta hookup, the 60Hz component of the real and reactive powers must be used to compute a total three-phase phase angle. The 60Hz component of the reactive power, VATOT[1] is computed using (20) with k = 1 (since 60Hz is the 1st harmonic), giving:
The 60Hz component of the real power, WTOT[1], can be obtained in an analogous fashion using:
This results in the following expression for θTOT:
4.5 Power Factors
Power factor and displacement power factor are computed with (11) and (12), with the use of WTOT, VATOT, and θTOT instead of the single phase W, VA, and θ.
5. Four Wire Delta
With a four wire delta circuit, individual phase powers and power factors cannot be computed without imposing assumptions such as a balanced load, balanced source, etc. The recorder only computes total quantities in this mode. These values are computed and recorded as channel one. These computations happen once per cycle, as in the wye case. The fourth channel is treated as an extra single phase channel with power calculations as detailed in Section 2. Real and reactive power are calculated using the three-wattmeter method, which uses all three voltage and current channels. The recorder itself is connected as a wye, with each voltage channel measuring from phase to neutral.
5.1 Total Powers
Real and reactive total power is computed as the sum of the individual channels’ real and reactive powers, computed as if they were part of a wye circuit. Thus, (14) and (15) can be used, with (3) and (10) used to compute channel powers as in the wye case. Total apparent power is computed with (21).
5.2 Phase Angle
The phase angle is computed with (25). To compute the 60Hz real and reactive power used in (24), all three voltage and current channels are utilized, as per the three-wattmeter methodology. The expressions for WTOT[1] and VARTOT[1] become:
and
6. 2.5 Element Wye
In a 2.5 element wye, the circuit is a four wire wye, but only two voltage channels are available for measurement (typically due to only having two monitoring PTs). Unlike a 3-wire delta, v1[n] + v2[n] + v3[n] = 0 no longer holds, and it’s not possible to make exact measurements. The best compromise is to create an artificial voltage channel from the vector sum of the two available channels. With the PMI standard 2.5 element hookup, phase B (channel two) is the synthetic channel:
v2[n] = –(v1[n] + v3[n])
If the voltages are balanced, this does give the correct reading. All other calculations proceed as in a normal four wire wye circuit, using the synthetic voltage channel two as needed.
The synthetic channel two voltage is not recorded itself. Instead, PMI recorders also measure and record the actual voltage on the channel two input; that input is available for any monitoring, even though it’s not used in the succeeding calculations.
