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Effects of transformer models on the voltage variation of an unbalanced distribution system in the presence of harmonic distortion
Electric Power Systems Research, 17 {1989) 199 - 207
199
Effects of Transformer Models on the Voltage Variation of an Unbalanced Distribution System in the Presence of Harmonic Distortion ELHAM B. MAKRAM, REBEKAH L. THOMPSON and ADLY A. GIRGIS
Electrical and Computer Engineering Department, Clemson University, Clemson, SC 29634-0918 (U.S.A.) {Received May 30, 1989)
ABSTRACT
Distribution transformer models at high frequency become a subject o f great concern in harmonic propagation studies in distribution and transmission systems owing to their impact on harmonic distortion. A laboratory based transformer model was recently investigated using a c o m p u t e r controlled harmonic generator. This paper presents the effects o f this transformer model on the voltage variation as functions o f the frequency o f an unbalanced distribution system in the presence o f harmonic distortion. The results are compared with the other available transformer models. A 44 k V distribution system is used as a test case and the voltage variation at each bus in the system is obtained using the developed c o m p u t e r program and the McGraw-Edison harmonic analysis program (V-HARM).
INTRODUCTION
Imbalance and harmonic distortion are present in distribution systems. Imbalance in p o w e r system networks m a y be caused by unbalanced loads, a single c o n d u c t o r outage, or unbalanced feeder configurations. Harmonic distortion in distribution systems can be caused by the increased use of SCR controlled motors, fluorescent lighting, photovoltaic generation, and other harmonic producing p h e n o m e n a [1, 2]. A single- or a three-phase transformer presents a c o m p o n e n t of major interest with regard to harmonic distortion. It is essential to develop accurate transformer models and to study their effect on the power networks [3]. Accurate modeling of transformers must account for the variation of the transformer parameters at
each single frequency in the system [4, 5]. The values of the resistance and inductance of a distribution transformer depend on the type of transformer and the relative magnitudes of the harmonic c o m p o n e n t s that make up the current and voltage waveforms [6]. It has n o t been known precisely h o w transformer resistance and inductance vary with frequency and transformer rating. According to ref. 2, the transformer resistance at 1500 Hz is approximately 1.8 times the 60 Hz resistance for a 10 kVA transformer. A McGraw-Edison study [7] states that the X / R ratio of a large distribution transformer remains constant with frequency and the transformer inductance can be assumed constant. This implies that the winding resistance at each harmonic increases by a factor equal to the harmonic order. It is obvious that the two studies in refs. 2 and 7 gave different values of transformer series resistance for a given frequency spectrum, as shown in Fig. 1. In addition, these t w o studies assumed constant transformer inductance with frequency. However, the two studies m a y have used t w o different types of transformers with different ratings, c o n d u c t o r size and construction. Little or no attention has been paid to the variation of the transformer inductance and the core losses at different frequencies in the system. A recent investigation has been reported on a laboratory experiment for transformer modeling in the presence of harmonic distortion using a c o m p u t e r controlled harmonic generator on a single- and three-phase transformer [8]. This study used the hybrid set of measurements to include the variation of the short-circuit and open-circuit parameters as functions of frequency. This paper describes the effects of the transformer model developed by the laboratory experiment and the other available Elsevier Sequoia/Printed in The Netherlands
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FREQUENCY Fig. 1. A comparison between the two transformer models: +, V-HARM transformer mode] (McGraw-Edison) [7 ]; ", H A R M flow transformer mode] (EPRI) [2 ].
models on the voltage variation at different buses in a power system as a function of harmonic order. The results are compared with some other available transformer models using a developed computer program and the McGraw-Edison harmonic analysis program.
THREE--PHASE
ADMITTANCE MATRIX
ALGORITHM
The three-phase admittance matrix Yb~ algorithm is of a general form so that it can be applied to any system. In this algorithm, each element is represented by a 3 X 3 submatrix of phase quantities abc. The submatrices are built according to the model representing each element such as feeders, loads, capacitors, and transformers [9, 10]. The diagonal submatrix of the Yb~ matrix for bus r is f o u n d by summing the 3 X 3 submatrices of the elements connected to this bus. For an N-bus system, the relation is expressed as
[Ybus]r,~ = ~ [Yabc]i,~ i=l
(1)
where [Yab¢]i, r is the three-phase admittance submatrix representing the element between bus i and bus r. The off-diagonal submatrices of the Ybus matrix can be calculated as [Ybus]r, j = --
[Yab¢] r, j
(2)
If there is no element between bus r and bus j, then this term is zero [10]. The details of the Ybus building algorithm are reported in ref. 9. Note that each element in the Ybus matrix is a function of frequency. For each new frequency, a new Ybus matrix must be built in order to consider the effects of the frequency dependent parameters. For an N-bus system the resulting Yb,~ matrix has a dimension of 3N × 3N. Once the Yb-~ matrix is computed, its inverse is obtained. This inverse is known as a three-phase bus impedance matrix Z b u s. Z b u s can also be obtained directly using the threephase bus impedance matrix building algorithm [9]. Next, a current vector is defined which contains the magnitudes and angles of the injected current source at bus m. The voltage variation at each bus due to this injected current source only is simply a product of the Zbu ~ matrix and the current vector:
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N o te t h a t all entries of the cur r ent vector are zero e x c e p t f o r t he 3 × 1 subvector which represents the bus at which the harmonic source is injected. Instead of multiplying the Zbus matrix by a ve c t or containing m any zeros, only the Zbu s elements which correspond to the non-zero current values are stored and used in the voltage calculations. Consider a cu r r e nt injected at a bus m with the three phases d e n o t e d as m, m + 1, and m + 2, as shown in eqn. (3). The phase voltages at bus r are expressed as
t
Y,os MATRIX l
+ MULTIPLY THE IMPEDANCE MATRIX BY THE CURRENT VECTOR TO OBTAIN THE VOLTAGE VECTOR
Fig. 2. F l o w c h a r t of t h e d e v e l o p e d h a r m o n i c analysis program.
V~ = [Zbu~]r ' mira where Vr is the 3 × 1 subvector representing voltages on phases r, r + 1, and r + 2, [Zbu~]r, m t h e three-phase impedance submatrix relating the voltage at bus r to the c ur r e nt at bus m, and lm the 3 × 1 subvector representing injected currents at phases m, m + 1, and m+2. T h e above p r o cedur e is repeated f or each harmonic f r e q u e n c y o f interest. A flowchart o f the developed harmonic analysis program is shown in Fig. 2.
EXAMPLE SYSTEM
The example system shown in Fig. 3 is used in a study o f the effects of the available t r a n s f o r m e r models on the voltage variations at d i f f er en t buses. T he system data are shown in Tables 1 - 4. It is also used in a comparison b e t w een the developed algorithm and the V-HARM program. This system is a 44 kV distribution system which serves mainly
TABLE 1 T r a n s m i s s i o n line d a t a From bus
To bus
kV ResisGeom. base t a n c e mean ( ~ / m i l e ) radius (ft)
Radius Length (ft) (miles)
44 k V bus 44 k V bus Cmn. Cmn. 44 k V bus Bus 4 Bus 6
Bus 1
44
0.278
0 . 0 2 4 4 0.03
0.19
Cmn.
44
0.196
0.029
0.0355
4.04
Bus 2 44 Bus 3 44 Bus 4 4 4
0.278 0.278 0.44
0 . 0 2 4 4 0.03 0.0244 0.03 0.0125 0.0173
7.96 0.56 2.26
Bus 6 44 Bus 7 44
0.278 0.278
0 . 0 2 4 4 0.03 0.0244 0.03
11.64 1.99
industrial customers. A 1250 hp DC drive provides a harmonic c u r r e n t source for t he system. This DC drive is m odel ed as a sixpulse variable speed drive rectifier. Harmonic c u r r e n t characteristics for this t y p e of
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1250 HP dc drives Fig. 3. 44 kV distribution system. TABLE 2
TABLE 4
Load data
Transformer data
Bus name
kV base
MVA rating
Power factor
Surry # 1 Bus 2L Bus 3L Bus 4L Surry # 6 Bus 7L
12.47 13.0 12.47 12.47 12.47 4.16
14567.5 6107.0 3047.0 3050.0 848.5 2089.5
98.3 95.5 79.7 95.6 79.3 85.0
TABLE 3 Capacitor data Bus name
kV base
Capacitor voltage
MVAR
Bus 2L Bus 3L Rutledge Bus 4L
13.0 12.47 44.0 12.47
13.8 13.8 46.0 13.8
1.05 1.8 6.0 0.75
From bus
44 kV bus
To bus
Transmission system Bus 1 Surry #1 Bus 2 Bus 2L Bus 3 Bus 3L Bus 4 Bus 4L Bus 6 Surry # 6 Bus 7 Bus 7L Bus 7L Drive A Bus 7L Drive B
Prim./sec. voltage (kV)
MVA
Impedance (%)
100]44
20.0
8.08
44112.47 44113 44112.47 44/12.47 44/12.47 44/4.16 4.16]0.6 4.16]0.6
0.5 7.5 5.0 0.75 0.5 7.5 2.5 2.5
5.5 6.5 6.55 4.91 5.0 4.6 4.58 4.58
r e c t i f i e r u s e d in t h e c o m p a r i s o n a r e g i v e n in T a b l e 5. T h e s y s t e m b a s e is 1 0 0 M V A in all studies.
203
TABLE 5 H a r m o n i c c u r r e n t c h a r a c t e r i s t i c s f o r a six-pulse variable s p e e d drive rectifier Harmonic order
C u r r e n t (%)
1 5 7 11 13 17 19 23 25
100 20 14 8 7 4.5 4 3 3
E F F E C T O F T H E T R A N S F O R M E R M O D E L ON T H E V O L T A G E V A R I A T I O N AT D I F F E R E N T BUSES
For this study, four transformer models are considered. The first is a test model devel-
oped by the authors and described in detail in ref. 8. The best-estimate functions of the parameters for the model are shown in Table 6. This model considers the core of the transformer by using open-circuit parameters along with short-circuit parameters. The second model is the EPRI [2] model which represents the series resistance as a function of the square of the frequency while the series inductance remains constant. The third model is the McGraw-Edison model [11] which presents the resistance as a linear function of frequency while the inductance remains constant. Finally, the fourth model is the CIGRE model [1] which is composed of a series frequency dependent resistance and reactance. In these four models, a one per unit current magnitude (96 225 A) is used at 60 Hz. The estimated functions for the short-circuit parameters in the above four models are shown in Table 7.
TABLE 6 Single-phase t r a n s f o r m e r test results Element
Best-estimate function
R~(~)
3.769066--0.2760622
L~(H)
2.725 × 10 -3 -- 4.694 × l O - 6 f + 9.075 X l O - 9 f 2 -- 6.579 × 10-12f 3 + 1.464 × l O - I S f s -- 1.894 X lO-2Sf 7
Roc(~)
- - 1 . 2 6 8 + 0 . 0 4 5 1 f - - 3.090 X l O - S f 2 + 5.166 × 1 0 - 9 f 3 + 7.936 X 10-16f s
Loc(H)
1.490 × 10 -2 -- 4.275 X l O - S f + 5.173 × l O - S f 2 -- 2.42 × l O - l l f 3 + 2.044 × l O - I S f s
X 1 0 - 3 f + 0.1543 × 1 0 - s f 2
Rsc = s h o r t - c i r c u i t resistance; Lsc = s h o r t - c i r c u i t inductance; f = frequency.
i n d u c t a n c e ; Roc = o p e n - c i r c u i t resistance; Loc = o p e n - c i r c u i t
TABLE 7 E s t i m a t e d f u n c t i o n s o f t h e s h o r t - c i r c u i t p a r a m e t e r s o f t h e available m o d e l s Model name
Estimated function R e s i s t a n c e (p.u.)
I n d u c t a n c e (p.u.)
Test
1.0029--7.3458
x 1 0 - s f + 4.1058 x 1 0 - 7 f 2
EPRI
0 . 9 9 5 3 + 6.0013 X 1 0 - s f + 2.9475 x 1 0 - 7 f 2
1.0
McGraw-Edison [11]
9 . 5 3 7 0 x 10 - 7 + 0 . 0 1 6 7 f
1.0
CIGRE
1 . 2 0 7 2 - - 0 . 0 0 6 5 f + 5.0674 X 1 0 - s f 2
1 . 0 0 0 6 - - 5.4019 × l O - 6 f -- 7.1592 x l O - S f 2
0 . 0 0 2 7 3 - - 4.69 x 1 0 - 6 f + 9.075 x 1 0 - 9 f 2 - - 6.579 x 1 0 - 1 2 f 3 + 1.464 X 1 0 - 1 s f s - - 1.894 x 1 0 - 2 s f 7
[2]
[1]
204 RESULTS
obtained using the EPRI model, the McGrawEdison model, and the CIGRE model differ from the test model results by 22.3%, 16% and 33.7%, respectively. For the second comparison, only the shortcircuit parameters are present in the test model. Figures 5(a) and (b) show the results of this study at bus 4L and bus 2L, respectively. Similar results to the previous comparison are found. Thus, the open-circuit parameters in the test model have only a small increasing effect on the voltage variation. The results of the V-HARM [11] and the developed program are then compared for the system in Fig. 3. Voltage variations at bus 7L and bus 4L are chosen for this comparison. Figures 6(a) and (b) show the voltage varia-
AND COMPARISON
For the first comparison, both the opencircuit and the short-circuit parameters are considered in the test model. Figures 4(a) and (b) show the results of the four transformer models at bus 4L and bus 2L, respectively. All models agree at the lower harmonics with variations beginning at the seventh harmonic. At bus 4L, the greatest differences occur at the 23rd harmonic. A difference of 38.66% exists between the test model results and the EPRI model results. The McGraw-Edison model results differ by 52% and the CIGRE model results by 67.5% from the test model results. At bus 2L, a significant difference is noted at the l l t h harmonic. The results
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tions of phase A at bus 7L and bus 4L, respectively. At bus 7L the V-HARM program gives results which are 6.73% to 13.8% less than the results from the developed program. At bus 4L there is a 3.18% difference between the results of the two programs at the 3rd harmonic and a 6.5% difference at the 13th harmonic. Figure 7 shows the results of the developed program using two different transformer models (test model and McGraw-Edison model). At bus 7L the greatest difference of 2.55% occurs at the 19th harmonic. At bus 4L the 25th harmonic gives the largest difference of 12.64%. Finally, the transformer test model is replaced by the McGraw-Edison linear transformer model in the developed program. This gives each program the same transformer
model. The resulting voltage variations of phase A at the two buses (bus 7L and bus 4L) in both programs are almost the same.
CONCLUSION
This paper presented a three-phase admittance matrix algorithm to be used in calculating voltage variations due to harmonic current injections. The impact of the transformer model on harmonic propagation is also investigated and reported. When the developed algorithm was used with the different transformer models, similar results for the voltage variations were obtained from the test model and the EPRI model {nonlinear model). In this comparison,
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the test m o d e l was used b o t h with and witho u t the o p e n - c i r c u i t p a r a m e t e r s . T h e s t u d y indicated that the open-circuit parameters have m i n o r e f f e c t s on t h e voltage variation in this p a r t i c u l a r t r a n s f o r m e r . T h e voltage variations o b t a i n e d f r o m t h e d e v e l o p e d a l g o r i t h m using t h e linear V - H A R M t r a n s f o r m e r m o d e l and t h e V - H A R M p r o g r a m were in a g r e e m e n t . When t h e linear V - H A R M t r a n s f o r m e r m o d e l was c o m p a r e d with the test m o d e l using t h e d e v e l o p e d algorithm, a d i f f e r e n c e in the results was n o t i c e a b l e . H o w e v e r , the a u t h o r s d o n o t claim t h e m o d e l s r e p r e s e n t e d here t o be applicable t o all t r a n s f o r m e r sizes. T h e p r o c e d u r e , con-
cepts, and testing m e t h o d o l o g y can be applied t o d i f f e r e n t sizes o f t r a n s f o r m e r s , if p r o p e r testing e q u i p m e n t is available. Finally, t r a n s f o r m e r s are n o n l i n e a r devices and n e e d t o be c o n s i d e r e d as such in harm o n i c analysis programs. T h e e f f e c t o f the core in the d i s t r i b u t i o n t r a n s f o r m e r m o d e l s s h o u l d be i n c l u d e d specially f o r large transformers.
REFERENCES 1 J. Arrillaga, D. A. Bradley and P. S. Bodger, Power System Harmonics, Wiley, 1985.
207
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Fig. 7. Comparison between two transformer models using the developed program: voltage variation of phase A at (a) bus 7 L and (b) bus 4L. - - , test model; . . . . . , McGraw-Edison model.
2 M. S. Hwang, W. M. Grady and H. W. Sanders, Jr., Distribution transformer winding losses due to nonsinusoidal currents, IEEE Trans., PWRD-2 (1987) 140 - 146. 3 A. A. Girgis and B. D. Guy, Computer-based data acquisition for teaching transients and switching phenomena and performing research on digital protection, . I E E E Trans., PWRS-3 (4) (1988) 1361 - 1368. 4 A. Keyhani, S. M. Miri and S. HaD, Parameter estimation for power transformer models from time-domain data, IEEE Trans., PWRD-1 (3) (1986) 140 - 146. 5 V. Brandswajn, H. W. Dommel and I. I. Dommel, Matrix representation of three-phase N-winding transformers for steady-state and transient studies, IEEE Trans., PAS-101 (1982) 1369 - 1378. 6 J. Avili-Rosales and F. L. Alvarado, Nonlinear frequency dependent transformer model for electromagnetic transient studies in power systems, IEEE Trans., PAS-101 (1982) 4 2 8 1 4288.
7 M. F. McGranaghan, R. C. Dugan, J. A. King and W. T. Jewell, Distribution feeder harmonic study methodology, IEEE Trans., PAS-103 (1984) 3663 - 3671. 8 E. B. Makram, R. L. Thompson and A. A. Girgis, A new laboratory experiment for transformer modeling in the presence of harmonic distortion using a computer controlled harmonic generator, IEEE Trans., PWRS-3 (4) (1988) 1857 - 1863. 9 E. B. Makram and A. A. Girgis, A generalized technique for the development of the three-phase impedance matrix for unbalanced power systems, Electr. Power Syst. Res., 15 (1988) 41 - 50. 10 G. T. Heydt, Computer Analysis Methods for Power Systems, Macmillan, New York, 1986, 11 E. W. Gunther and M. F. McGranaghan, A PC-
Based Simulation Program for Power S y s t e m Harmonic Analysis, McGraw-Edison Power Systems Division, Cooper Industries, Inc., 1988; obtainable from Thomas Edison Tech. Center, Franksville, WI 53126-9526.
199
Effects of Transformer Models on the Voltage Variation of an Unbalanced Distribution System in the Presence of Harmonic Distortion ELHAM B. MAKRAM, REBEKAH L. THOMPSON and ADLY A. GIRGIS
Electrical and Computer Engineering Department, Clemson University, Clemson, SC 29634-0918 (U.S.A.) {Received May 30, 1989)
ABSTRACT
Distribution transformer models at high frequency become a subject o f great concern in harmonic propagation studies in distribution and transmission systems owing to their impact on harmonic distortion. A laboratory based transformer model was recently investigated using a c o m p u t e r controlled harmonic generator. This paper presents the effects o f this transformer model on the voltage variation as functions o f the frequency o f an unbalanced distribution system in the presence o f harmonic distortion. The results are compared with the other available transformer models. A 44 k V distribution system is used as a test case and the voltage variation at each bus in the system is obtained using the developed c o m p u t e r program and the McGraw-Edison harmonic analysis program (V-HARM).
INTRODUCTION
Imbalance and harmonic distortion are present in distribution systems. Imbalance in p o w e r system networks m a y be caused by unbalanced loads, a single c o n d u c t o r outage, or unbalanced feeder configurations. Harmonic distortion in distribution systems can be caused by the increased use of SCR controlled motors, fluorescent lighting, photovoltaic generation, and other harmonic producing p h e n o m e n a [1, 2]. A single- or a three-phase transformer presents a c o m p o n e n t of major interest with regard to harmonic distortion. It is essential to develop accurate transformer models and to study their effect on the power networks [3]. Accurate modeling of transformers must account for the variation of the transformer parameters at
each single frequency in the system [4, 5]. The values of the resistance and inductance of a distribution transformer depend on the type of transformer and the relative magnitudes of the harmonic c o m p o n e n t s that make up the current and voltage waveforms [6]. It has n o t been known precisely h o w transformer resistance and inductance vary with frequency and transformer rating. According to ref. 2, the transformer resistance at 1500 Hz is approximately 1.8 times the 60 Hz resistance for a 10 kVA transformer. A McGraw-Edison study [7] states that the X / R ratio of a large distribution transformer remains constant with frequency and the transformer inductance can be assumed constant. This implies that the winding resistance at each harmonic increases by a factor equal to the harmonic order. It is obvious that the two studies in refs. 2 and 7 gave different values of transformer series resistance for a given frequency spectrum, as shown in Fig. 1. In addition, these t w o studies assumed constant transformer inductance with frequency. However, the two studies m a y have used t w o different types of transformers with different ratings, c o n d u c t o r size and construction. Little or no attention has been paid to the variation of the transformer inductance and the core losses at different frequencies in the system. A recent investigation has been reported on a laboratory experiment for transformer modeling in the presence of harmonic distortion using a c o m p u t e r controlled harmonic generator on a single- and three-phase transformer [8]. This study used the hybrid set of measurements to include the variation of the short-circuit and open-circuit parameters as functions of frequency. This paper describes the effects of the transformer model developed by the laboratory experiment and the other available Elsevier Sequoia/Printed in The Netherlands
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FREQUENCY Fig. 1. A comparison between the two transformer models: +, V-HARM transformer mode] (McGraw-Edison) [7 ]; ", H A R M flow transformer mode] (EPRI) [2 ].
models on the voltage variation at different buses in a power system as a function of harmonic order. The results are compared with some other available transformer models using a developed computer program and the McGraw-Edison harmonic analysis program.
THREE--PHASE
ADMITTANCE MATRIX
ALGORITHM
The three-phase admittance matrix Yb~ algorithm is of a general form so that it can be applied to any system. In this algorithm, each element is represented by a 3 X 3 submatrix of phase quantities abc. The submatrices are built according to the model representing each element such as feeders, loads, capacitors, and transformers [9, 10]. The diagonal submatrix of the Yb~ matrix for bus r is f o u n d by summing the 3 X 3 submatrices of the elements connected to this bus. For an N-bus system, the relation is expressed as
[Ybus]r,~ = ~ [Yabc]i,~ i=l
(1)
where [Yab¢]i, r is the three-phase admittance submatrix representing the element between bus i and bus r. The off-diagonal submatrices of the Ybus matrix can be calculated as [Ybus]r, j = --
[Yab¢] r, j
(2)
If there is no element between bus r and bus j, then this term is zero [10]. The details of the Ybus building algorithm are reported in ref. 9. Note that each element in the Ybus matrix is a function of frequency. For each new frequency, a new Ybus matrix must be built in order to consider the effects of the frequency dependent parameters. For an N-bus system the resulting Yb,~ matrix has a dimension of 3N × 3N. Once the Yb-~ matrix is computed, its inverse is obtained. This inverse is known as a three-phase bus impedance matrix Z b u s. Z b u s can also be obtained directly using the threephase bus impedance matrix building algorithm [9]. Next, a current vector is defined which contains the magnitudes and angles of the injected current source at bus m. The voltage variation at each bus due to this injected current source only is simply a product of the Zbu ~ matrix and the current vector:
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N o te t h a t all entries of the cur r ent vector are zero e x c e p t f o r t he 3 × 1 subvector which represents the bus at which the harmonic source is injected. Instead of multiplying the Zbus matrix by a ve c t or containing m any zeros, only the Zbu s elements which correspond to the non-zero current values are stored and used in the voltage calculations. Consider a cu r r e nt injected at a bus m with the three phases d e n o t e d as m, m + 1, and m + 2, as shown in eqn. (3). The phase voltages at bus r are expressed as
t
Y,os MATRIX l
+ MULTIPLY THE IMPEDANCE MATRIX BY THE CURRENT VECTOR TO OBTAIN THE VOLTAGE VECTOR
Fig. 2. F l o w c h a r t of t h e d e v e l o p e d h a r m o n i c analysis program.
V~ = [Zbu~]r ' mira where Vr is the 3 × 1 subvector representing voltages on phases r, r + 1, and r + 2, [Zbu~]r, m t h e three-phase impedance submatrix relating the voltage at bus r to the c ur r e nt at bus m, and lm the 3 × 1 subvector representing injected currents at phases m, m + 1, and m+2. T h e above p r o cedur e is repeated f or each harmonic f r e q u e n c y o f interest. A flowchart o f the developed harmonic analysis program is shown in Fig. 2.
EXAMPLE SYSTEM
The example system shown in Fig. 3 is used in a study o f the effects of the available t r a n s f o r m e r models on the voltage variations at d i f f er en t buses. T he system data are shown in Tables 1 - 4. It is also used in a comparison b e t w een the developed algorithm and the V-HARM program. This system is a 44 kV distribution system which serves mainly
TABLE 1 T r a n s m i s s i o n line d a t a From bus
To bus
kV ResisGeom. base t a n c e mean ( ~ / m i l e ) radius (ft)
Radius Length (ft) (miles)
44 k V bus 44 k V bus Cmn. Cmn. 44 k V bus Bus 4 Bus 6
Bus 1
44
0.278
0 . 0 2 4 4 0.03
0.19
Cmn.
44
0.196
0.029
0.0355
4.04
Bus 2 44 Bus 3 44 Bus 4 4 4
0.278 0.278 0.44
0 . 0 2 4 4 0.03 0.0244 0.03 0.0125 0.0173
7.96 0.56 2.26
Bus 6 44 Bus 7 44
0.278 0.278
0 . 0 2 4 4 0.03 0.0244 0.03
11.64 1.99
industrial customers. A 1250 hp DC drive provides a harmonic c u r r e n t source for t he system. This DC drive is m odel ed as a sixpulse variable speed drive rectifier. Harmonic c u r r e n t characteristics for this t y p e of
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1250 HP dc drives Fig. 3. 44 kV distribution system. TABLE 2
TABLE 4
Load data
Transformer data
Bus name
kV base
MVA rating
Power factor
Surry # 1 Bus 2L Bus 3L Bus 4L Surry # 6 Bus 7L
12.47 13.0 12.47 12.47 12.47 4.16
14567.5 6107.0 3047.0 3050.0 848.5 2089.5
98.3 95.5 79.7 95.6 79.3 85.0
TABLE 3 Capacitor data Bus name
kV base
Capacitor voltage
MVAR
Bus 2L Bus 3L Rutledge Bus 4L
13.0 12.47 44.0 12.47
13.8 13.8 46.0 13.8
1.05 1.8 6.0 0.75
From bus
44 kV bus
To bus
Transmission system Bus 1 Surry #1 Bus 2 Bus 2L Bus 3 Bus 3L Bus 4 Bus 4L Bus 6 Surry # 6 Bus 7 Bus 7L Bus 7L Drive A Bus 7L Drive B
Prim./sec. voltage (kV)
MVA
Impedance (%)
100]44
20.0
8.08
44112.47 44113 44112.47 44/12.47 44/12.47 44/4.16 4.16]0.6 4.16]0.6
0.5 7.5 5.0 0.75 0.5 7.5 2.5 2.5
5.5 6.5 6.55 4.91 5.0 4.6 4.58 4.58
r e c t i f i e r u s e d in t h e c o m p a r i s o n a r e g i v e n in T a b l e 5. T h e s y s t e m b a s e is 1 0 0 M V A in all studies.
203
TABLE 5 H a r m o n i c c u r r e n t c h a r a c t e r i s t i c s f o r a six-pulse variable s p e e d drive rectifier Harmonic order
C u r r e n t (%)
1 5 7 11 13 17 19 23 25
100 20 14 8 7 4.5 4 3 3
E F F E C T O F T H E T R A N S F O R M E R M O D E L ON T H E V O L T A G E V A R I A T I O N AT D I F F E R E N T BUSES
For this study, four transformer models are considered. The first is a test model devel-
oped by the authors and described in detail in ref. 8. The best-estimate functions of the parameters for the model are shown in Table 6. This model considers the core of the transformer by using open-circuit parameters along with short-circuit parameters. The second model is the EPRI [2] model which represents the series resistance as a function of the square of the frequency while the series inductance remains constant. The third model is the McGraw-Edison model [11] which presents the resistance as a linear function of frequency while the inductance remains constant. Finally, the fourth model is the CIGRE model [1] which is composed of a series frequency dependent resistance and reactance. In these four models, a one per unit current magnitude (96 225 A) is used at 60 Hz. The estimated functions for the short-circuit parameters in the above four models are shown in Table 7.
TABLE 6 Single-phase t r a n s f o r m e r test results Element
Best-estimate function
R~(~)
3.769066--0.2760622
L~(H)
2.725 × 10 -3 -- 4.694 × l O - 6 f + 9.075 X l O - 9 f 2 -- 6.579 × 10-12f 3 + 1.464 × l O - I S f s -- 1.894 X lO-2Sf 7
Roc(~)
- - 1 . 2 6 8 + 0 . 0 4 5 1 f - - 3.090 X l O - S f 2 + 5.166 × 1 0 - 9 f 3 + 7.936 X 10-16f s
Loc(H)
1.490 × 10 -2 -- 4.275 X l O - S f + 5.173 × l O - S f 2 -- 2.42 × l O - l l f 3 + 2.044 × l O - I S f s
X 1 0 - 3 f + 0.1543 × 1 0 - s f 2
Rsc = s h o r t - c i r c u i t resistance; Lsc = s h o r t - c i r c u i t inductance; f = frequency.
i n d u c t a n c e ; Roc = o p e n - c i r c u i t resistance; Loc = o p e n - c i r c u i t
TABLE 7 E s t i m a t e d f u n c t i o n s o f t h e s h o r t - c i r c u i t p a r a m e t e r s o f t h e available m o d e l s Model name
Estimated function R e s i s t a n c e (p.u.)
I n d u c t a n c e (p.u.)
Test
1.0029--7.3458
x 1 0 - s f + 4.1058 x 1 0 - 7 f 2
EPRI
0 . 9 9 5 3 + 6.0013 X 1 0 - s f + 2.9475 x 1 0 - 7 f 2
1.0
McGraw-Edison [11]
9 . 5 3 7 0 x 10 - 7 + 0 . 0 1 6 7 f
1.0
CIGRE
1 . 2 0 7 2 - - 0 . 0 0 6 5 f + 5.0674 X 1 0 - s f 2
1 . 0 0 0 6 - - 5.4019 × l O - 6 f -- 7.1592 x l O - S f 2
0 . 0 0 2 7 3 - - 4.69 x 1 0 - 6 f + 9.075 x 1 0 - 9 f 2 - - 6.579 x 1 0 - 1 2 f 3 + 1.464 X 1 0 - 1 s f s - - 1.894 x 1 0 - 2 s f 7
[2]
[1]
204 RESULTS
obtained using the EPRI model, the McGrawEdison model, and the CIGRE model differ from the test model results by 22.3%, 16% and 33.7%, respectively. For the second comparison, only the shortcircuit parameters are present in the test model. Figures 5(a) and (b) show the results of this study at bus 4L and bus 2L, respectively. Similar results to the previous comparison are found. Thus, the open-circuit parameters in the test model have only a small increasing effect on the voltage variation. The results of the V-HARM [11] and the developed program are then compared for the system in Fig. 3. Voltage variations at bus 7L and bus 4L are chosen for this comparison. Figures 6(a) and (b) show the voltage varia-
AND COMPARISON
For the first comparison, both the opencircuit and the short-circuit parameters are considered in the test model. Figures 4(a) and (b) show the results of the four transformer models at bus 4L and bus 2L, respectively. All models agree at the lower harmonics with variations beginning at the seventh harmonic. At bus 4L, the greatest differences occur at the 23rd harmonic. A difference of 38.66% exists between the test model results and the EPRI model results. The McGraw-Edison model results differ by 52% and the CIGRE model results by 67.5% from the test model results. At bus 2L, a significant difference is noted at the l l t h harmonic. The results
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tions of phase A at bus 7L and bus 4L, respectively. At bus 7L the V-HARM program gives results which are 6.73% to 13.8% less than the results from the developed program. At bus 4L there is a 3.18% difference between the results of the two programs at the 3rd harmonic and a 6.5% difference at the 13th harmonic. Figure 7 shows the results of the developed program using two different transformer models (test model and McGraw-Edison model). At bus 7L the greatest difference of 2.55% occurs at the 19th harmonic. At bus 4L the 25th harmonic gives the largest difference of 12.64%. Finally, the transformer test model is replaced by the McGraw-Edison linear transformer model in the developed program. This gives each program the same transformer
model. The resulting voltage variations of phase A at the two buses (bus 7L and bus 4L) in both programs are almost the same.
CONCLUSION
This paper presented a three-phase admittance matrix algorithm to be used in calculating voltage variations due to harmonic current injections. The impact of the transformer model on harmonic propagation is also investigated and reported. When the developed algorithm was used with the different transformer models, similar results for the voltage variations were obtained from the test model and the EPRI model {nonlinear model). In this comparison,
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the test m o d e l was used b o t h with and witho u t the o p e n - c i r c u i t p a r a m e t e r s . T h e s t u d y indicated that the open-circuit parameters have m i n o r e f f e c t s on t h e voltage variation in this p a r t i c u l a r t r a n s f o r m e r . T h e voltage variations o b t a i n e d f r o m t h e d e v e l o p e d a l g o r i t h m using t h e linear V - H A R M t r a n s f o r m e r m o d e l and t h e V - H A R M p r o g r a m were in a g r e e m e n t . When t h e linear V - H A R M t r a n s f o r m e r m o d e l was c o m p a r e d with the test m o d e l using t h e d e v e l o p e d algorithm, a d i f f e r e n c e in the results was n o t i c e a b l e . H o w e v e r , the a u t h o r s d o n o t claim t h e m o d e l s r e p r e s e n t e d here t o be applicable t o all t r a n s f o r m e r sizes. T h e p r o c e d u r e , con-
cepts, and testing m e t h o d o l o g y can be applied t o d i f f e r e n t sizes o f t r a n s f o r m e r s , if p r o p e r testing e q u i p m e n t is available. Finally, t r a n s f o r m e r s are n o n l i n e a r devices and n e e d t o be c o n s i d e r e d as such in harm o n i c analysis programs. T h e e f f e c t o f the core in the d i s t r i b u t i o n t r a n s f o r m e r m o d e l s s h o u l d be i n c l u d e d specially f o r large transformers.
REFERENCES 1 J. Arrillaga, D. A. Bradley and P. S. Bodger, Power System Harmonics, Wiley, 1985.
207
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Fig. 7. Comparison between two transformer models using the developed program: voltage variation of phase A at (a) bus 7 L and (b) bus 4L. - - , test model; . . . . . , McGraw-Edison model.
2 M. S. Hwang, W. M. Grady and H. W. Sanders, Jr., Distribution transformer winding losses due to nonsinusoidal currents, IEEE Trans., PWRD-2 (1987) 140 - 146. 3 A. A. Girgis and B. D. Guy, Computer-based data acquisition for teaching transients and switching phenomena and performing research on digital protection, . I E E E Trans., PWRS-3 (4) (1988) 1361 - 1368. 4 A. Keyhani, S. M. Miri and S. HaD, Parameter estimation for power transformer models from time-domain data, IEEE Trans., PWRD-1 (3) (1986) 140 - 146. 5 V. Brandswajn, H. W. Dommel and I. I. Dommel, Matrix representation of three-phase N-winding transformers for steady-state and transient studies, IEEE Trans., PAS-101 (1982) 1369 - 1378. 6 J. Avili-Rosales and F. L. Alvarado, Nonlinear frequency dependent transformer model for electromagnetic transient studies in power systems, IEEE Trans., PAS-101 (1982) 4 2 8 1 4288.
7 M. F. McGranaghan, R. C. Dugan, J. A. King and W. T. Jewell, Distribution feeder harmonic study methodology, IEEE Trans., PAS-103 (1984) 3663 - 3671. 8 E. B. Makram, R. L. Thompson and A. A. Girgis, A new laboratory experiment for transformer modeling in the presence of harmonic distortion using a computer controlled harmonic generator, IEEE Trans., PWRS-3 (4) (1988) 1857 - 1863. 9 E. B. Makram and A. A. Girgis, A generalized technique for the development of the three-phase impedance matrix for unbalanced power systems, Electr. Power Syst. Res., 15 (1988) 41 - 50. 10 G. T. Heydt, Computer Analysis Methods for Power Systems, Macmillan, New York, 1986, 11 E. W. Gunther and M. F. McGranaghan, A PC-
Based Simulation Program for Power S y s t e m Harmonic Analysis, McGraw-Edison Power Systems Division, Cooper Industries, Inc., 1988; obtainable from Thomas Edison Tech. Center, Franksville, WI 53126-9526.