Derating of Anisotropic Transformers under Nonsinusoidal Operating Conditions

Electrical Power and Energy Systems 25 (2003) 1±12

www.elsevier.com/locate/ijepes

Derating of Anisotropic Transformers under Nonsinusoidal Operating Conditions M.A.S. Masoum a,*, E.F. Fuchs b a

b

Department of Electrical Engineering, Iran University of Science and Technology, Tehran, Iran Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309-0425, USA Received 12 December 2000; revised 3 July 2001; accepted 29 August 2001

Abstract An iterative algorithm is developed for the computation of transformer derating factors (functions) under (non)sinusoidal excitation and linear (R, RL, RC) loads. The proposed nonlinear model is a combination of frequency domain (to incorporate harmonics) and time domain (to account for saturation) techniques and only requires transformer parameters and l±i characteristics. The derating factors are generated using the computed transformer apparent powers under `rated losses' and `rated fundamental output voltage' conditions. Computed voltage and current waveforms and total losses are compared with measurements and show good agreements. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Transformers; Derating; Harmonics; Power quality

1. Introduction Transformer derating is considered as a protection technique in highly distorted power systems. For the computation of derating factors, an accurate nonlinear transformer model must be available. Anisotropic nonlinear transformers exhibit several nonlinearities, which complicate their analysis [1±5]: (1) the saturation behavior of the nonlinear magnetic core, (2) the anisotropy of with- and cross-grain laminations, (3) the dependency of iron-core losses on maximum value of the fundamental ¯ux density, and (4) their dependency on the harmonic components (amplitude and phase shift) of induced voltage (¯ux density). The ®rst two nonlinearities have been dealt with in the calculation of l±i characteristics [6]. The latter two nonlinearities have been considered in the calculation of the loss±density characteristics for the yokes and legs of the transformer [7]. To account for the fourth nonlinearity, so called phase-factor functions are de®ned and measured which re¯ect the increase or decrease in the iron-core losses caused by terminal voltage harmonics [7]. It is the goal of this paper to use the l±i and loss±density functions of Refs. [6,7] in an iterative large-signal nonlinear transformer model to compute the additional transformer * Corresponding author. E-mail address: [email protected] (M.A.S. Masoum).

losses (compared with the rated losses) and derating factors under nonsinusoidal conditions. To con®rm the validity of computed results, some of them are compared with measurements for the single-phase transformer with following name plate data: High voltage side (secondary): 240/480 V Low voltage side (primary): 120/240 V KVA: 1.00; Frequency: 50/60 Hz.

2. Transformer nonlinear model For the calculation of single valued ¯ux linkage curves [6] and no-load losses [7] of anisotropic single-phase transformers, instantaneous magnetic ®eld solutions are required. These solutions consist of the vector potential at each node of the grid systems employed to ®nd the B±H characteristics of the yokes and the legs of the transformer and the ¯ux density and reluctivity in each mesh. In addition, loss± density characteristics for the yokes and the legs of the transformer, phase-factor functions for low-order harmonics and the ¯ux density versus ¯ux linkage function for each individual mesh are available [7: Figs. 3, 7 and 4, respectively]. These characteristics will be used to compute the losses and transformer responses under various loads and excitations.

0142-0615/03/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0142-061 5(01)00091-6

2

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

Fig. 1. Equivalent circuit of a transformer.

Using the equivalent circuit of anisotropic transformers (Fig. 1), we have:   i h …h† …h† ZP…h† EP…h† ˆ VP…h† 2 Imag 1 Icore

l…h† P ˆ

EP…h† jhv1

ZS…h†

0

ZP…h† 1 ZS…h†

0

for h ˆ 1; 3; 5; ¼ 0

…1†

…2† 0

…h† …h† …h† 0 where ZP…h† ˆ R…h† P 1 jvhLP ; ZS ˆ RS 1 jvhL S and EP ; …h† …h† …h† …h† VP ; Imag ; Icore ; lP are the hth harmonic components of the induced and terminal voltages, magnetizing and core-loss …h† 0 currents and ¯ux linkage, respectively. R…h† P and RS are the transformer primary and secondary resistances at h(60)Hz taking into account skin effect. In order to compute the losses under different load and terminal voltage conditions, the iterative algorithm of Fig. 2 is used. For LP and L 0S ; the computed nonlinear primary and secondary inductance of Ref. [6] are used. The required l±i; l±B and phase-factor functions for the desired transformer can be computed using Refs. [6,7]. If these characteristics are not available, one could use the measured l±i characteristics, assuming negligible core-losses. In that case, l±B and phase-factor functions are set to zero and core-losses are not computed. The algorithm is as follows (Fig. 2): Step 1: Fourier analyze the input voltage generating voltage harmonics VP…h† : Assume zero initial values for har…h† …h† monic core-loss …Icore † and magnetizing …Imag † currents. Step 2: Find the harmonics of induced voltage and ¯uxlinkage by employing Eqs. (1) and (2). Underrelax the magnitude of EP…h† by de®ning the new value as uEP…h† unew ˆ uEP…h† un and the old value as uEP…h† uold ˆ uEP…h† un21 ; then



analyze it to generate the harmonics of the magnetizing …h† †: current …Imag Step 4: Use the ¯ux-linkage functions, the loss±density …l±B† characteristics and the phase-factor functions to compute the harmonic core losses …P…h† core † as demonstrated in Ref. [7]. Compute core-loss resistances and core-loss currents.



 R…h† core ˆ

0

IS…h† ˆ

ZS…h†

…5†

for h ˆ 1; 3; 5; ¼

0

0

h ˆ 1; 3; ¼

…6†

…7†

If core-loss current is not required or l±B and/or phasefactor functions are not available, skip this step. Step 6: Compute harmonic copper losses as P…h† copper ˆ

  1  …h† 2 …h†  …h† 0 2 …h† 0 I P RP 1 I S RS 2

…8†

Step 7: Total losses could be assumed as the sum of copper and core losses, or may be computed using input and output powers,

Ploss ˆ

2

uEP…h† uold

EP…h†

…h† …h† IP…h† ˆ IS…h† 1 Imag 1 Icore

where URF is the underrelaxation factor. Step 3: Synthesize the instantaneous ¯ux-linkage time function …l…t†† and use the l±i characteristics to compute the instantaneous magnetizing current …imag …t††: Fourier

1 URF

uEP…h† unew

…4†

2Pcore …h†

Step 5: Calculate transformer currents as

…3†

ˆ

uEP…h† uold

2

h i …h† …h† 1=2 uIcore u ˆ P…h† core =Rcore

Ploss ˆ

uEP…h† un

EP…h†

X hˆ1;3;5;¼

X hˆ1;3;5;¼

h

…h† P…h† copper 1 Pcore

i

…9A†

   i 0 1 h …h† …h† …h† 0 …h† uVP u´uIP ucos f…h† 2 uV…h† P S u´uIS ucos fS 2

…9B†

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

3

Fig. 2. The iterative algorithm for computing transformer input and output signals and copper and core losses under saturation and/or nonsinusoidal condition.

…h† where f…h† P and fS are the phase shifts between the hth harmonic of the primary and secondary voltages and currents, respectively. Step 8: Check whether solution has converged. If convergence is insuf®cient, repeat Steps 1±8. One way of monitoring convergence is to observe the change in total losses

DPloss ˆ

Ploss un 2 Ploss un21 £ 100% Ploss un21

…10†

where n is the iteration number. If DPloss is below 1%, solution has suf®ciently converged. If Eq. (9A) is used, convergence is achieved after about ®fteen iterations …n ù 15†: If Eq. (9B) is used, convergence

and f…h† must be deteriorates (e.g. n ù 40) since f…h† P S computed very accurately.

3. Additional loss calculation The input terminal voltage is assumed to contain a certain amplitude and phase shift of a low-order harmonic. Two distinct phase shifts of the imposed terminal voltage harmonics are considered; a phase shift of 08 where the peak-to-peak value of the input voltage is maximum (peaky waveform), and a phase shift of 1808 where the peak-to-peak value of the input voltage is minimum (¯at waveform). Five types of loads are selected: rated resistive

4

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

To achieve rated output voltage, the iterative method of Fig. 2 is used, but the input terminal voltage is adjusted to make the fundamental of secondary voltage identical to its rated value (Fig. 3). VP…h† unew

ˆ

VP…h† uold

Rated VS…1† VS…1†

0

0

h ˆ 1; 3; ¼

…11†

the percentage increase of losses is %DP ˆ

Fig. 3. The iterative algorithm for computing percentage increase of copper and core losses under saturation and/or nonsinusoidal conditions.

(R 0L ˆ 65 V), rated parallel RC with power factors of 0.8 (R 0L ˆ 81:3 V, C 0L ˆ 25:5 mF) and 0.9 (R 0L ˆ 72:3 V, C 0L ˆ 17:8 mF) and rated series RL with power factors of 0.8 (R 0L ˆ 52 V, L 0L ˆ 103:5 mH) and 0.9 (R 0L ˆ 58:5V, L 0L ˆ 75:2 mH).

Ploss 2 Rated loss £ 100 Rated loss

…12†

where `Ploss' is the sum of copper and iron-core losses for nonsinusoidal excitation under rated output conditions. Figs. 4±7 show some of the calculated input and output waveforms. Percentage increase of losses and the calculated R…h† core are listed in Tables 1 and 2 and are also plotted in Figs. 8 and 9. Note that for a power factor of 0.8, input terminal voltages containing a 9th harmonic generate near resonance conditions, since leakage inductance [6, Fig. 22] and load p capacitance result in f0 ˆ 1=…2p LC 0L † ˆ 530:5 Hz. The same condition holds for the 11th harmonic and a power factor of 0.9 where f0 ˆ 635 Hz.

Fig. 4. Input terminal voltage containing 10% of 7th harmonic with a phase shift of f…7† ˆ 08 at resistive load: (a) input terminal voltage, calculated output voltage, primary and secondary (referred to primary) currents and ¯ux linkages, (b) calculated primary, magnetizing and core-loss currents.

Fig. 5. Input terminal voltage containing 10% of 7th harmonic with a phase shift of f…7† ˆ 1808 at resistive load: (a) input terminal voltage, calculated output voltage, primary and secondary (referred to primary) currents and ¯ux linkages, (b) calculated primary, magnetizing and core-loss currents.

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

5

Table 1 Calculated percentage increases of losses and core-losses resistances for nonsinusoidal input terminal voltages as compared with sinusoidal excitation for resistive load h

1 3 5 7 9 11 3 5 7 9 11

VP…h† =VP…1† (%)

Sin 5 5 5 5 5 20 20 20 20 20

f…h† ˆ 08

f…h† ˆ 1808

DP…h† loss (%)

R…1† c (V)

R c…h† (V)

…h† DP loss (%)

R…1† c (V)

R c…h† (V)

± 0.40 0.28 0.02 0.70 0.15 1.30 5.90 1.90 8.50 4.40

2112 2116 2107 2118 2085 2111 2156 2057 2212 1926 2110

±

Rated loss ˆ 76:7 W 0.28 0.08 0.40 2 0.06 0.14 6.10 2.20 7.70 3.50 4.90

2108 2114 2100 2122 2112 2081 2158 1979 2142 2080

± ± ± ± ± 1224 4721 2163 2226 1970

622

± ± ± ± 50,310 1553 2501 1938 1880

Table 2 Calculated percentage increases of losses and core-loss resistances for nonsinusoidal terminal voltages as compared with sinusoidal excitation for parallel capacitive load with p.f. ˆ 0.9 h

1 3 5 7 9 11 3 5 7 9 11

VP…h† =VP…1† (%)

Sin 5 5 5 5 5 20 20 20 20 20

f…h† ˆ 08

f…h† ˆ 1808

DP…h† loss (%)

R…1† c (V)

R c…h† (V)

…h† DP loss (%)

R…1† c (V)

R c…h† (V)

±

2120 2125 2115 2127 2088 2120 2167 2057 2243 1903 2109

±

Rated loss ˆ 74:3 W 0.31 1.20 3.87 11.40 22.10 9.50 19.20 62.20 186 355

2116 2123 2106 2127 2120 2087 2175 1963 2129 2099

± ± ± ± ± 1224 5284 2828 3736 1667

0.78 1.20 3.40 12.80 22.20 6.30 22.80 55.90 192 355

± ±

638

524 ± 43,709 1659 3173 3187 1674

Fig. 6. Input terminal voltage containing 10% of 7th harmonic with a phase shift of f…7† ˆ 08 at parallel RC load with p.f. ˆ 0.9: (a) input terminal voltage, calculated output voltage, primary and secondary (referred to primary) currents and ¯ux linkages, (b) calculated primary, magnetizing and core-loss currents.

6

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

Fig. 7. Input terminal voltage containing 10% of 7th harmonic with a phase shift of f…7† ˆ 1808 at parallel RC load with p.f. ˆ 0.9: (a) input terminal voltage, calculated output voltage, primary and secondary (referred to primary) currents and ¯ux linkages, (b) calculated primary, magnetizing and core-loss currents.

Fig. 8. Percentage increases of losses at rated resistive load due to single-harmonic voltage amplitudes of 5, 10, 15, and 20%.

Fig. 9. Percentage increases of losses at parallel RC load with p.f. ˆ 0.9 due to single-harmonic voltage amplitudes of 5, 10, 15, and 20%.

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

4. Computing derating functions For the computation of derating functions and the derating of transformers under nonsinusoidal operating regimes, the total losses and the fundamental amplitude of the output voltage must be at rated values. The derating factor in per unit (DF) is: h ih i …1† …h† …h† DF ˆ S…1† = S …13† u u S V ±0 S V ˆ0 P

P

…1† where S…1† S uVP…h† ±0 and SS uVP…h† ˆ0 are the secondary fundamental apparent powers at nonsinusoidal and sinusoidal inputs, respectively. To achieve rated losses and fundamental output voltage, the iterative method of Fig. 10 is applied. Starting with rated input terminal voltage, the solution proceeds from pri-

7

mary to the secondary (forward solution), determining the total losses …Ploss † and output voltage …VS…1† †: Thereafter, alternately adjusting either the input harmonic voltage amplitudes 0

VP…h† unew

VS…1† ˆ V …h† u Rated VS P new

…14†

or the magnitudes of the load impedance 0

VL…h† unew ˆ

Rated losses …h† 0 ZL uold Ploss

…15†

4.1. Computing initial condition The above method of calculating the derating factor

Fig. 10. The iterative algorithm for computing transformer derating functions (factors) under saturation and/or nonsinusoidal conditions.

8

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

Table 3 Computed derating functions (factors) at rated fundamental output voltage and losses (resistive load) VP…h† =VP…1† (%)

h

3 5 7 9 11 3 5 7 9 11 3 5 7 9 11 3 5 7 9 11

5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

Derating factor

f…h† ˆ 08

f…h† ˆ 1808

1.000 0.999 1.000 0.996 0.998 1.000 0.989 0.996 0.975 0.985 0.996 0.975 0.990 0.954 0.985 0.991 0.955 0.987 0.928 0.963

0.997 1.000 0.998 0.999 0.998 0.987 0.999 0.984 0.990 0.985 0.975 0.993 0.968 0.982 0.973 0.959 0.990 0.948 0.974 0.961

3 5 7 9 11 3 5 7 9 11 3 5 7 9 11 3 5 7 9 11

4.2. The derating algorithm (Fig. 10) The iterative procedure is as follows (Steps 1±3 are for computing initial condition and could be skipped at the cost of slow convergence): Step 1: Input the nonsinusoidal terminal voltage. Assume rated fundamental terminal voltage. Set the ¯ag (¯ag ˆ on). Step 2 (Forward Solution): Use the algorithm of Fig. 2 to compute losses and fundamental output voltage. Step 3 (Backward Solution): Step 3.1: Adjust the harmonic amplitudes of the output current as: s Rated losses …h† 0 IS uold ˆ Ploss

…16†

Step 3.2: Adjust the magnitude of load impedance such that rated fundamental amplitude of the output voltage is obtained for current of Step 3.1 0

ZL…h† unew ˆ

Rated VS…1† …h† 0 ZL uold : Ploss ZL…1†

VP…h† =VP…1† (%)

h

(where the solution proceeds in forward direction) converges. However, to improve convergence, the initial condition of the primary voltage is estimated through a solution proceeding from the secondary to the primary (backward solution).

0 IS…h† unew

Table 4 Computed derating functions (factors) at rated fundamental output voltage and losses (parallel RC load with p.f. ˆ 0.9)

…17†

Step 3.3: Compute the induced voltage harmonics,

5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

assuming EP…h† ˆ ES…h† 0

EP…h† ˆ IS…h† ZS…h† 0

Derating factor

f…h† ˆ 08

f…h† ˆ 1808

1.000 0.991 0.972 0.910 0.816 0.989 0.956 0.902 0.773 0.641 0.972 0.908 0.822 0.664 0.542 0.952 0.851 0.754 0.579 0.475

0.997 0.993 0.967 0.917 0.815 0.982 0.963 0.891 0.784 0.641 0.962 0.823 0.804 0.681 0.541 0.935 0.877 0.723 0.602 0.473

0

0

…18† 0

0

…h† 0 where ZS…h† ˆ R…h† S 1 jvhLS 1 ZL : Step 3.4: Use the algorithm of Fig. 2 to compute magnetizing and core-loss currents. Step 3.5: The estimated input voltage harmonics for the initial condition is  0  …h† …h† VP…h† ˆ EP…h† 1 IS…h† 1 Imag ZP…h† 1 Icore …19†

where ZP…h† ˆ R…h† P 1 jvhLP : Step 4 (Forward Solution): With the input voltage and load impedance of Step 3, compute Ploss and VS…1† (using Fig. 2). Step 5 (Achieving Derating Conditions): Step 5.1: If ¯ag ˆ on, then use Eq. (14) to adjust the input voltage and set ¯ag ˆ off. Step 5.2: If ¯ag ˆ off, then use Eq. (15) to adjust the load impedance. Underrelax the change in load impedance DZunew ˆ DZuold 1 URF…DZunew 2 DZuold †

…20†

where URF ˆ 1; except for 9th and 11th harmonics at capacitive loads use URF ˆ 0:2: Set ¯ag ˆ on. Step 6 (Check for Convergence): If total losses are close to the rated losses (e.g. within 0.2%) and the fundamental amplitude of the output voltage is approaching the rated value (e.g. within 0.1%) the solution is converged, otherwise repeat Steps 4±6. The errors are de®ned as

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12 Table 5 Computed derating functions (factors) at rated fundamental output voltage and losses (series RL load with p.f. ˆ 0.9) h

3 5 7 9 11 3 5 7 9 11 3 5 7 9 11 3 5 7 9 11

VP…h† =VP…1† (%)

5 5 5 5 5 10 10 10 10 10 15 15 15 15 15 20 20 20 20 20

Derating factor

f…h† ˆ 08

f…h† ˆ 1808

1.000 0.996 0.999 0.992 0.996 1.000 0.990 0.998 0.978 0.988 1.000 0.981 0.999 0.959 0.983 1.000 0.966 1.000 0.944 0.982

0.993 1.000 0.995 0.997 0.995 0.985 1.000 0.987 0.994 0.988 0.975 1.000 0.976 0.992 0.982 0.962 1.000 0.961 0.994 0.979

Ploss 2 Rated losses £ 100% Rated losses

Computed derating functions (factors) are shown in Tables 3±5 for resistive, capacitive and inductive loads, where low-order harmonics (e.g. 3rd, 5th, 9th, 11th) have amplitudes of 5 and 20%.tpb 12

5. Experimental veri®cation To support the method of calculation as presented in this paper, some of the computed results are veri®ed by measurements. An adjustable phase-lock circuit drives a power ampli®er such that any harmonic with a selected amplitude and phase shift (with respect to the fundamental at 60 Hz) can be superimposed with the fundamental of selected amplitude. Experiments are performed for waveshape con®gurations where voltage harmonics of about 20% amplitudes and phase shifts of about f…h† ˆ 08 and about f…h† ˆ 1808 are superimposed with the fundamental voltage. Two methods for measuring the thermal impact of harmonics on single-phase transformers are presented:

follows Perror ˆ

9

…21A†

1. Temperature measurementsÐwhere the losses are assumed to be proportional to the temperature rise of the transformer and are measured using thermocouples. 2. Power measurementsÐwhere the losses are calculated using measured nonsinusoidal input and output voltages and currents. The losses can be de®ned as

Verror ˆ

VS…1† 2 Rated VS…1† £ 100% Rated VS…1†

…21B†

Step 7 (Computing Derating Factor): Compute derating factors (Eq. (13)) for different nonsinusoidal input conditions. The procedure of Fig. 10 converges in about eight iterations. However, for the 9th and 11th voltage harmonics at capacitive loads …URF ˆ 0:2† the number of iterations increases to 15 since the losses are very high for such loads (near resonance).

Ploss ˆ

1 ZT ‰v …t†i …t† 2 vS …t†iS …t†Šdt T 0 P P

…22†

Unfortunately, Eq. (22) yields only accurate values (error ,3%) for losses if the primary and secondary voltages and currents are measured with very high accuracy (error ,0.05%). For transformers with transformation ratios of about one, a better approach for measuring losses under nonsinusoidal regimes consists of two measurements, where ‰vS …t† 2 vP …t†Š and ‰iS …t† 1 iP …t†Š as well as ‰vS …t† 1 vP …t†Š and

Fig. 11. Input voltage containing about 20% of 5th harmonic with f…h† < 08 at resistive load: (a) given terminal voltage, measured and calculated primary currents, (b) measured and calculated secondary voltages and currents.

10

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

Fig. 12. Input voltage containing about 20% of 5th harmonic with f…h† < 1808 at resistive load: (a) given terminal voltage, measured and calculated primary currents, (b) measured and calculated secondary voltages and currents.

Fig. 13. Input voltage containing about 20% of 5th harmonic with f…h† < 08 at parallel RC load with p.f. ˆ 0.92: (a) given terminal voltage, measured and calculated primary currents, (b) measured and calculated secondary voltages and currents.

‰iS …t† 2 iP …t†Š are measured simultaneously. With the loss components p1 …t† ˆ ‰vS …t† 2 vP …t†Š‰iS …t† 1 iP …t†Š and p2 …t† ˆ ‰vS …t† 1 vP …t†Š‰iS …t† 2 iP …t†Š; the total losses become 1 ZT Ploss ˆ 2 ‰p …t† 1 p2 …t†Šdt …23† 2T 0 1 The advantage of Eq. (23) as compared to Eq. (22) is that the order of the magnitude of error for the losses are about the same as those for primary and secondary voltage sum (difference) and current difference (sum). The above methods of measuring losses was used to measure transformer primary and secondary voltage and

current waveforms and losses for about 20% single-harmonic amplitudes of terminal voltage and about rated output apparent power at resistive (R 0L ˆ 65:03 V) and parallel RC (R 0L ˆ 64:4; C 0L ˆ 13:4mF) loads. Some calculated and measured waveforms are compared in Figs. 11±14. Calculated and measured losses are compared in Table 6 (columns 6 and 7) for about 20% of 3rd and 5th harmonics for resistive as well as RC loads. No measurements are performed for the 7th, 9th and 11th harmonics since the total losses are extremely high at capacitive loads. Table 6 (columns 8±10) illustrates the temperature

Fig. 14. Input voltage containing about 20% of 5th harmonic with f…h† < 1808 at parallel RC load with p.f. ˆ 0.92: (a) given terminal voltage, measured and calculated primary currents, (b) measured and calculated secondary voltages and currents.

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

11

Table 6 Calculated and measured iron-core and copper losses at rated resistive (R 0L ˆ 65:03 V) and parallel RC (R 0L ˆ 69:4 V, C 0L ˆ 13:4 mF, p.f. ˆ 0.92) loads Power measurements and calculation error (%) VP…1†

(V)

h

VP…h† =VP…1†

(%)

Type

Resistive load (R 0L ˆ 65:03 V) 377.4 sin ± ± 368.1 3 21 Max a 372.9 3 20.3 Min b 369.9 5 20.9 Max 374.0 5 19.8 Min RC load (R 0L ˆ 69:4 V, C 0L ˆ 13:4 mF, p.f. ˆ 0.92) 367.1 Sin ± ± 372.1 3 20.2 Max 376.0 3 20.5 Min 365.0 5 21.1 Max 375.0 5 19.9 Min a b

Temperature measurements …h†

f

(8)

Pcal loss

Pmeas loss

(W)

(W)

T …h† (8)

…h† Tamb (8)

DT …h† (%)

±

76.73 74.5 (22.9%) 80.3 (4.7%) 78.9 (2.7%) 78.1 (1.8%)

76.6 80.2 (4.6%) 80.8 (5.4%) 81.4 (6.2%) 80.5 (5.1%)

85.3 85.5 89.7 88.5 87.2

23.3 24.1 23.2 23.7 23.5

± 21.0 7.3 4.5 2.7

±

72.16 77.8 (7.8%) 82.5 (14.4%) 83.0 (15.0%) 84.5 (17.1%)

71.5 80.7 (12.8%) 79.8 (11.5%) 85.6 (19.7%) 84.4 (18.1%)

78.2 81.6 86.9 88.8 87.2

23.1 24.1 23.4 23.3 23.6

±

3 2185 22.3 2159 24 2179 23 2160

4.3 15.3 18.9 15.4

Peak-to-peak value of terminal voltage in about maximum (peaky waveform). Peak-to-peak value of terminal voltage in about minimum (¯at waveform).

obtained for rated operation at about rated resistive and parallel capacitive loads for about 20% harmonic amplitude at phase angles of about f…h† ˆ 08 and f…h† ˆ 1808:

3.

6. Discussion and conclusions A large-signal iterative model for anisotropic nonlinear transformers [6,7] is used to determine the additional losses and ensuing derating factors (functions) as well as the derating of transformers due to single-harmonic excitations. The saturation of the iron-core, the anisotropy of with- and cross-grain laminations, the dependency of the iron-core losses on the amplitudes and phase shifts of harmonics leads to surprising, however, physically plausible results which are con®rmed by temperature and power loss (Table 6) measurements. The main results are: 1. At rated apparent output power operation (e.g. 1 kV A), the rated losses for sinusoidal input terminal voltage are lowest for capacitive and highest for inductive loads. The reason for this phenomenon lies in the amplitude of the ¯ux linkage time function (Figs. 4 and 7) being lowest for capacitive operating regimes. 2. For low-order single-harmonic excitations, the harmonic phase shift f…h† plays an important role in the generation of additional harmonic losses. There exist nonsinusoidal waveshapes of the input terminal voltage, where at rated fundamental apparent output power, the total losses are about the same or even less (Tables 1±6) than the rated losses generated under sinusoidal input terminal voltage and rated fundamental apparent output power. The reason for this fortuitous situation is that the iron-core losses depend on the ¯ux linkage waveshape (that is the harmonic shift) and the reduction of the iron-core losses due to the reduced ¯ux linkage amplitude can be larger than the additional harmonic ohmic losses. This phenom-

4.

5. 6.

enon is greatly dependent upon the lamination thickness, the saturation level of the iron core and the relative magnitudes of the iron-core and ohmic losses. The additional losses due to a single-harmonic excitation are lower for inductive and highest for capacitive loads (Tables 1 and 2 and Figs. 8±12). The results for resistive loads con®rm the measurements as presented in Refs. [5,7]. In power systems very often power factor correcting capacitors are placed near transformers and it is possible that the additional harmonic losses become excessive (Fig. 9) resulting in overheating or even failure of transformers. Derating functions (Tables 3±5) specify the reduced fundamental apparent output power of transformers under nonsinusoidal operation with one single harmonic, while the output voltage and the losses are maintained at rated values. It is realized that in practice there never exists one single harmonic only within the input terminal voltage, however, there is mostly one single dominant harmonic. This paper shows that the derating of transformers for resistive and inductive loads are small (1% for 5% harmonic voltage amplitudes, see Tables 3 and 5). For capacitive loads, however, the derating can be large (20% for 5% voltage harmonic, see Table 4). The fundamental and harmonic core-loss resistances vary widely within large bounds. The carefully performed measurements yield total losses and temperature rises that con®rm computed results with good accuracy.

References [1] Elleuch M, Poloujadoff M. New transformer model including joint air gaps and lamination anisotropy. IEEE Trans Magn 1998;3452:3701±11. [2] Pierrat L, Tran-Quoc T, Montmeat A. Nonliear transient simulation of transformers. Conf Rec IEEE 1995;2:1193±8. [3] Vakilian M, Degeneff RC, Kupferschmid M. Computing the internal transient voltage respond of a transformer with a nonlinear core using

12

M.A.S. Masoum, E.F. Fuchs / Electrical Power and Energy Systems 25 (2003) 1±12

Gear's method, Part I: veri®cation. IEEE Trans Power Deliv 1995;102:702±8. [4] Vakilian M, Degeneff RC, Kupferschmid M. Computing the internal transient voltage respond of a transformer with a nonlinear core using Gear's method, Part II: theory. IEEE Trans Power Deliv 1995;104:1836±42. [5] Fuchs EF, You Y, Roesler DJ. Modeling and simulation, and their validation of three-phase transformers with three legs under DC bias. IEEE Trans Power Deliv 1999;142:443±9.

[6] Fuchs EF, Masoum MAS, Roesler DJ. Large signal nonlinear model of anisotropic transformers for nonsinusoidal operation, Part I: l±i characteristics. IEEE Trans Power Deliv 1990;6(1):174±85. [7] Masoum MAS, Fuchs EF, Roesler DJ. Large signal nonlinear model of anisotropic transformers for nonsinusoidal operation, Part II: magnetizing and core-loss currents. IEEE Trans Power Deliv 1991;6(4):1509±16.