1
Harmonics effect on rectifier current imbalance V. Gelman, Member, IEEE,
Abstract-- Analyzed data collected during the field test of close coupled 12 pulse rectifier-transformer with Wye and Delta bridges connected in parallel. Observed the Wye Delta current imbalance higher then expected based on transformer data. Found the substantial presence of 5th harmonic in the incoming line voltage. Provided explanation of measured difference between the Wye Delta voltages due to harmonics presence. Presented the effect of harmonics on the rectifier current imbalance. Calculated current imbalance based on the observed harmonic content and transformer during the tests.
During the test we also measured the rectifier transformer secondary voltages at no load and observed about 0.9% difference between Wye (higher) and Delta. It was an indication of harmonics effects. Oscillograms of incoming line
Index Terms-- Harmonic distortion, Rectifiers
I. INTRODUCTION
Wye and Delta voltages 200.00
150.00
100.00
50.00
3301
3151
3001
2851
2701
2551
2401
2251
2101
1801
1651
1501
1351
1201
901
1051
751
601
451
`
1951
delta
0.00 301
In modern power systems there might be a substantial harmonics presence due to proliferation of nonlinear loads such as power supplies for consumer electronics, computers, office equipment, etc. and also to the power electronic converters. Allowable harmonics content is govern by IEEE519 standard which establishes that no single harmonics should exceed 3% and total harmonic distortion (THD) should be below 5% Adhering to the IEEE519 standard assures reliable operation of power grid with no noticeable effects. However, there is one area where harmonics can provide measurable effect even if they remain below IEEE519 specified level – it is current unbalance in 12 pulse diode rectifiers.
voltage showed significant distortion (see Fig. 1). Two other traces are secondary voltages for Delta (R1-R3) and Wye (R2R4). The secondary voltages for Delta and Wye show different amplitudes because there were different measurement circuits. There were problems getting the digitized waveforms for secondary voltages from the equipment, so we used mathematical conversion of line voltage to get the secondary voltages (namely, took the digitized line voltage, shifted it by 1/3 of its period, then subtracted the original and shifted
1
II. HARMONICS EFFECT
Fig. 1 Primary and secondary voltages
151
The transformer-rectifier unit consists of a transformer with dual secondary windings (Wye and Delta), bus duct and 12pulse rectifier. There is a published theory of such rectifier operating from a power supply with sinusoidal voltage (see ref 1). The reference 1 deals with series connection of the rectifier bridges where the same current flows through both of them and there is no current unbalance. However, modern rectifiers have normally Wye and Delta bridges connected in parallel to reduce semiconductor losses and improve overall efficiency. In such case we have two voltage sources with different impedances operating in parallel. Any difference in the voltages and impedances will create current imbalance.
wye
-50.00
-100.00
A. Field observation We observed the current unbalance higher than calculated values during full voltage load tests at the traction substation. V. Gelman is with VG Controls, Inc, Vernon, NJ 07462 USA (e-mail:
[email protected]).
-150.00
-200.00
Fig. 2 Reconstructed secondary voltages
curves and finally divided result by current transfer ratio, 19/11). Figure 2 shows the reconstructed secondary voltages
2
for Delta and Wye windings, the reconstructed shape is very close to the original oscillograms. B. Waveform analysis The Fourier transform was performed on the digitized line voltage waveform, the resulting harmonics are shown in the Table 1. TABLE I HARMONIC CONTENT OF RECONSTRUCTED Y/DELTA SECONDARY
difference, which is very close to the calculated. As shown in the Appendix, we can expect the same difference (1%) between no-load voltages of Wye and Delta bridges. Wye harmonics 200
150
VOLTAGES 100
Average
first harm 5th harm
Delta harmonics 200.00
150.00
100.00 Line V first harm
3251
3126
3001
2876
2751
2626
2501
2376
2251
2126
2001
1876
1751
1626
1501
1376
1251
1126
876
751
1001
626
501
376
1
99.63%
251
0
The 5th harmonics has a very high value, about 2.5% of fundamental. As shown in the Appendix it can create substantial voltage difference between Wye and Delta rectifier groups. Figure 3 shows the original digitized incoming voltage trace together with its 1st and 5th harmonic. There is also 5th harmonic expanded 10 times for clarity. Since transformer has Delta primary, the Delta secondary voltage has the same shape as incoming voltage. From fig. 3 we can see that 5th harmonic is almost out of phase with the incoming voltage, actually the phase shift is 180-34=146 deg. Figure 4 shows the same harmonics for Wye secondary voltages From fig. 4 we can see that 5th harmonic is in phase with the first harmonic. As shown in the Appendix, this phase relationship increases the output of Wye bridge by 0.41% and at the same time decreases the output of Delta bridge by the same number. Since the transformer has current transfer ratio of 19/11 which is by 0.28% lower then ideal 3 , without voltage
50.00
Line V
50
108.84 126
Harmonic# 1 5 7 11 13 RMS Harmonic 109.25 2.76 0.34 0.34 0.21 109.29 voltages (V) Harmonic 100 % 2.52% 0.31% 0.31% 0.19% 100.03% voltages (%)
5th mult 10x
-50
-100
-150
-200
Fig.4 Wye secondary voltage harmonics
III. UNBALANCE IN 12 PULSE RECTIFIER WITH PARALLEL CONNECTION WYE/DELTA GROUPS The 12 pulse rectifier consists of two diode bridges connected in parallel. Each bridge is connected to its secondary transformer winding through the bus. Any unbalance between the transformer windings will cause unbalance between the rectifier sections. A transformer has two sources of current unbalance: difference between Wye and Delta winding voltages and difference in their impedances. Additional source of the current unbalance is the voltage distortion. The voltage difference between the winding is due to the fact that their ratio is determined by the number of turns, in our case 11 and 19. The ratio is 1.7272, it is by 0.28% smaller then 3. As shown above, the voltage distortion increases the noload voltage difference between Wye and Delta rectifiers from 0.28% to 1%. We can use the following formula for calculating current unbalance for the rectifier operating in linear Mode1 [see ref 2] where indices 1 and 2 correspond to Wye and Delta.
5th harm 3305
3187
3069
2951
2833
2715
2597
2479
2361
2243
2125
2007
1889
1771
1653
1535
1417
1299
1181
945
827
1063
709
591
473
355
1
237
119
0.00
5th mult 10x
-50.00
-100.00
-150.00
-200.00
Fig. 3 Incoming line voltage harmonics
distortion we would have seen the Wye voltage higher than Delta by 0.28%. Because of the 5th harmonic the difference measured by average AC voltmeter increases to 0.28+2*0.41=1.1%. Computer calculations for all harmonics gave slightly lower number of 1% or 5.4 VAC at 540 VAC secondary voltage. We actually measured 5 to 6 VAC
∆Id%
( x 2 − x1 )
(1) ∆Vline 1 Id% + x 2 + x1 2 ⋅ ∆Ed% Vline 2 x com + + 2R 2 Id1 − Id2 - difference between Delta and Wye DC = Idrated
∆Id% =
current relative to the rectifier rated current
x com - common part of transformer (and system) impedance,
x1 and x 2 - secondary part of the impedance,
including the buswork and rectifier itself. For our transformer, Xcom – is 5.32%, X1 and X2 are 1.5% and 0.8% (assuming 0.5% for buswork impedance) R - resistive part of the secondary impedance and buswork,
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transformer itself is about 0.3% ∆Ed% - voltage regulation of the rectifier, about 4% in our case Substituting the numbers in the equations gives
The (1) and (2) can be used only if absolute value of current unbalance ∆Id% is lower than the total rectifier current Id% . If it becomes higher, then we should use Id% or −Id% as a value for current unbalance. We can use this formula to calculate the unbalance with and without voltage distortion caused by harmonics. In the first case we have voltage differences to be 1%, in the second case it is 0.28%. We did not have instrumentation to measure DC currents in Wye and Delta groups directly, instead we used the secondary phase currents to estimate the DC currents. For the rectangular AC current waveshape with 120 deg conduction angle the relationship between DC and AC current is 3 IAC = KF ⋅ IAC = 1.225 ⋅ IAC 2
IDC =
(3)
However, under real life condition the coefficient will differ from 1.225 due to commutation angle and DC current ripples. If we assume that all phase currents have similar shape (i.e. similar ripples and commutation angles) then they have the same KF and we can estimate the DC currents as IR1 + IR3 + IR5 3 I +I +I = K F R2 R4 R6 3
IDCDelta = K F IDCWye
IDCDelta − IDC W ye IDCDelta + IDCWye
=
(5)
IR1 + IR3 + IR5 I −I −I − K F R2 R4 R6 I +I +I −I −I −I 3 3 = R1 R3 R5 R2 R4 R6 I +I +I I −I −I K F R1 R3 R5 + K F R2 R4 R6 IR1 + IR3 + IR5 + IR2 + IR4 + IR6 3 3
For the many aspects of the rectifier operation such as interphase transformer (IPT) condition or additional harmonics generation, it is important to know the difference between Wye and Delta currents
(6)
It is convenient to present it as a percentage of rated DC current. Iδ % =
IDCDelta − IDC W ye Idrated
=
67
117
718
849
873
1764
1771
R5 (A AC)
66
117
722
854
875
1772
1780
R2 (A AC)
238
603
1248
1369
1372
2173
2199
R4 (A AC)
237
609
1260
1382
1386
2196
2221
R6 (A AC)
240
612
1265
1388
1393
2204
2231
IDC (A DC)
306
806
2322
2630
2666
4726
4774
5
14
41
46
47
82
83
Mesrd IÄ - IY
-3.0
-9.5
-11.0
-10.9
-10.5
-8.9
-9.3
Calctd IÄ - IY Calctd IÄ - IY wo distortion
-5.3
-12.2
-10.8
-10.5
-10.5
-8.5
-8.5 -4.9
-3.4
-3.0
-1.6
-1.3
-1.2
0.7
0.7
IDC%
150
4.3
The chart on Figure 5 shows the results of these calculations. We can see that calculated current unbalance is close to the measured value if we take harmonic distortion into account. The lower unbalance at 14% load and below is due to the discontinuous operation of Delta group. Wye-Delta Current imbalance 15.0%
(0.2)
10.0%
KF
Iδ = IDCDelta − IDC W ye
R3 (A AC)
(4)
To get relative current unbalance ε DC =
TABLE 2 MEASURED AND CALCULATED CURRENT UNBALANCE TS00 TS037 TS037 TS003 TS003 TS003 TS003 150 Test# 1 008 006 005 004 001 000 % R1 (A AC) 67 117 717 848 871 1760 1766
IDCDelta − IDC W ye IDCDelta + IDC W ye Id ⋅ = ε DC = ε DCId % IDCDelta + IDC W ye Idrated Id rated
(7) Where: Id is total DC current, the sum of Wye and Delta currents, Idrated is rectifier rated current and Id% is rectifier loading. To measure AC currents we used Rogowski coil probes (AEMC Flexible current probe DK22935E-45-36 with dual range 300/3000A and corresponding scales of 10 mV/A and 1 mV/A). The oscillograms were taken with Yokogawa DL716 data acquisition system. We measured AC currents in all 6 secondary phases and also measured total rectifier combined current Id. The results are present in the Table 1. To calculate DC current unbalance we took the reading of all 6 phase currents (lines R1, R3, R5 – Delta; R2, R4,R6 – Wye) and them calculated DC unbalance
Delt-Wye imbalnce, % of rated
∆ Id% = 0.05 ⋅ Id%
(2)
∆Vline + 12.5 Vline
using an equations (5) and (7). Table 2 shows the measured data along with the calculated value from the equations both with and without the effect of harmonic voltages.
5.0%
(0.3)
0.0% 0%
20%
40%
60%
80%
100%
120%
140%
Measured Calculated w. harm. Calc. wo harm
-5.0%
-10.0%
-15.0% Rectifier Current Id (%)
(0.4) Fig 5. Measured and calculated unbalance IV. CONCLUSIONS The current unbalance between parallel Wye and Delta groups depends on both the harmonic voltages (both their amplitude and phase) and a mismatch between transformer current transfer ratios and impedances for the respective groups. The transformer parameters stay the same while voltage harmonics might change depending on other loads and changes in the external feeders connections. Harmonic voltage changes, either amplitude or phase, will change the current unbalance. Measured values are close to calculated current unbalance; calculations were based on the harmonics content extracted from the digitized incoming line voltage.
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V. APPENDIX A. Effect of voltage harmonics on the measured value of the voltage. There are two types of AC voltmeters used presently: so called true RMS and average value. The true RMS voltmeter measures the value of (8) 1 2 VRMS =
V T∫
In case of periodic signal it the same as VRMS =
∑
(9)
A k2 2
where Ak is the amplitude of kth harmonic. Under normal condition (IEEE-519 standard) the maximum value of each individual harmonics is below 3% of fundamental while their cumulative effect or total harmonic distortion (THD) is below 5%. Then measured value is 2 2 2 A (10) A A A ∑ k A ( THD ) VRMS =
1
2
∑ A
k
1
1 1 + k >1 2 = 1 1 + ≈ 2 ⋅ A1 2 2
2
∫ 2 2T
Lets estimate the effect of high order harmonic on the meter readings. Since the amplitude of high order harmonics are small, we can consider them individually. If more then one harmonics is present, combined effect will be equal the sum of effects from each harmonic. Lets assume that the voltage is (12) A1 sin ω t + A N sin (Nω t + ϕ ) st
th
where A1 and AN are the amplitudes of the 1 and N harmonics and ö is the phase of Nth harmonics. In order to calculate average rectified value of this voltage we assume that AN is a small portion of A1 and therefore the voltage sign is determined by the first harmonic. The average rectified value then becomes π
2π
∫( A sinτ + A sin( Nτ +ϕ) ) dτ −∫ ( A sinτ +A sin( Nτ +ϕ) ) dτ = 1
N
1
N
π
0
= 4A1 +
2AN
( cosϕ −cos( Nπ +ϕ) )
(13)
N
If we have even harmonic, then the cos (Nπ + ϕ ) = cos ϕ
N A1
If for example, we have the 5th harmonic with an amplitude of 5% of fundamental, we might get the reading change up to 1%. B. Wye Delta transformers and harmonics. Lets analyze the harmonics propagation through WyeDelta transformer If we have primary Delta voltages V1, V2 and V3 and a transformer with current transfer ratio of 3 then the output Wye voltage will be V − V3 (17) V = 1 2
3
Lets assume that there is only a single Nth harmonic is present and that (18) V = A sin τ + A sin (N τ + ϕ ) 1
1
N
2π 2π 2π 2π N τ + A N sin N τ − + ϕ = A 1 sin τ − τ + A N sin N τ + ϕ − V3 = A 1 sin τ − 3 3 3 3
Since THD is below 5% the difference between Vrms and the fundamental harmonic A1 is below 0.125% in the worst case. If the THD is below 3% (practical case), the difference is below 0.05%. So we can state that true RMS meter shows the RMS value of the fundamental harmonic. The average value meter rectifies the voltage and then measures its average value. The meter is calibrated to show correct RMS value for the sinusoidal signal. π 1 (11) V = V avg
and the relative change in average rectified meter reading is 1 AN (16) cos ϕ
(14)
and the effect of the harmonic on the meter reading is zero. However, for odd harmonics (15) cos (Nπ + ϕ ) = − cosϕ
V2 =
V1 − V3 3
=
A1 2π AN 2Nπ sinτ − sin τ − + sin ( Nτ + ϕ ) − sin Nτ + ϕ − 3 N 3 3 3
(19)
The first term is π π 2A1 sin τ + ⋅ sin 6 3 = A1 sinτ ∗ 3
where
τ∗ =τ +
π 6.
(20)
It shows that fundamental harmonic of Wye
voltage has the same amplitude as Delta and it is leading the Delta voltage by 30 degrees. The second term is Nπ Nπ π Nπ Nπ 2AN cos Nτ + ϕ − 2A N sin Nτ + ϕ + − ⋅ sin ⋅ sin 3 3 2 3 3 = 3 3 π ∗ or if we use τ = τ + 6 the term becomes π Nπ Nπ 2AN sin Nτ ∗ + ϕ + − ⋅ sin 2 2 3 3
(21)
(22)
In a real system we normally do not have even harmonics. For harmonics multiples of 3 (3, 9, 15, …) the term is zero – these harmonics will be cancelled by the Wye-Delta conversion. The remaining harmonics can be described by the formulae (23) N = 6k ± 1 They are 5, 7, 11, 13, etc. These harmonics are generated by all sorts of power conversion equipment, including rectifiers. For harmonics with N = 12k + 6 ± 1 such as 5, 7, 17, 19, etc the term becomes:
A N sin (Nτ ∗ + ϕ + π ) = − A N sin (Nτ ∗ + ϕ )
(24)
In other words, these harmonics have opposite effect on Wye and Delta voltages. For harmonics with N = 12k ± 1 such as 11, 13, 23, 25, etc the term is (25) A N sin Nτ ∗ + ϕ
(
)
or those harmonics have the same effect on Wye and Delta voltages.
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C. Harmonics effect on rectifier no-load voltage We can estimate effect of harmonic voltages on the rectifier using the same method as for the effect on the average AC meter reading. We will assume that we have only one harmonic present and that is small, so it does not affect diode commutations. To calculate no-load voltage of 6-pulse rectifier we need to integrate the line voltage from π to 2π . Assuming that line 3
3
voltage is V1 = A1 sinτ + A N sin (Nτ + ϕ )
(26)
we get Ed0
3 = π
2π 3
∫ (A
π 3
1
sinτ + A N sin (Nτ + ϕ ) ) dτ
(27)
the first terms integrates as 3 A 1 π
while the second term is 3 AN π 2π +ϕ cos N + ϕ − cos N π N 3 3
(28)
As explained above, we are interested only in harmonics with orders N = 6k ± 1. For these, the expression above evaluates to 3 AN (29) cos ϕ π N
In other words, relative effect of harmonics on the 6-pulse rectifier no-load voltage is ∆Ed0 1 A N (30) = cos ϕ Ed0
N A1
the same formulae as for the measured average type AC voltmeter (16). The effect of harmonic voltages on the rectifier at different current levels depends on the commutation angle and delay angle (if the rectifier operates as part of 12 pulse system. Generally it will be less then is ∆E 1 A N (31) = Ed0
N A1
D. Harmonics effect on rectifier voltage at arbitrary load The effect of harmonics on the rectifier voltage will vary with the rectifier current due to: 1. more complex waveform due to commutation interval in mode 1, and 2. changing phase shift of the harmonic voltage with respect of rectifier conduction interval due to commutation delay in mode 2 (see ref 2) The resulting formula are more complex and will be published later E. Effect of transformer coupling on the voltage unbalance due to harmonics In the appendix we did not use transformer coupling data. The results are applicable to both loose and closely coupled transformers operating in mode 1 and 2 as defined in [ref 2].
These modes cover the load range from 0% to over 500% for typical case. VI. REFERENCES [1]
[2]
[3]
R. L. Witzke, J. V. Kresser, J. K. Dillard “Influence of A-C Reactance on Voltage Regulation of 6 phase Rectifiers”, AIEE Transaction, vol 72 Jul 1953, pp 244-253 R. L. Witzke, J. V. Kresser, J. K. Dillard “Voltage regulation of 12Phase Double-Way Rectifiers”, AIEE Transaction, vol 72 Nov 1953, pp 689-697 C. Desoer, E. Kuh “Basic Circuit Theory” McGraw Hill, 1969
