Simulation of Random Variation of Three-phase Voltage Unbalance Resulting from Load Fluctuation Using Correlated Gaussian Random Variables

Transcription

1 Proc. Natl. Sci. Counc. ROC(A) Vol. 4, No. 3, 000. pp Simulation of Random Variation of Three-phase Voltage Unbalance Resulting from Load Fluctuation Using Correlated Gaussian Random Variables YAW-JUEN WANG Department of Electrical Engineering National Yun-Lin University of Science & Technology Yun-Lin, Taiwan, R.O.C. (Received January 1, 1999; Accepted November 10, 1999) ABSTRACT This paper deals with the modeling and simulation of random variation of three-phase voltage unbalance caused by load fluctuation in the distribution system. Active and reactive powers delivered by each phase to a distribution feeder are modeled by six correlated Gaussian random variables. Application of the Cholesky decomposition method allows the correlated Gaussian random variables to be effectively simulated using the Monte Carlo technique. This is followed by a simple iterative process for solving the load flow problem so that three-phase voltage phasors at the point of common coupling (PCC) can be obtained and random variation of voltage unbalance factor at the PCC can be evaluated. Key Words: voltage unbalance, Cholesky decomposition, Monte Carlo simulation, correlated Gaussian random variables I. Introduction Three-phase voltage imbalance is a frequently encountered disturbance in electric distribution systems. Imbalance of three-phase voltage results from asymmetry of line/cable impedances and from inequality of the loads in the three phases (Wang et al., 199). The former is related to the structure of the electric power system in which geometric allocation of lines/cables significantly influences their impedances. Efforts are in general made to reduce the asymmetry of the transmission line impedances by means of transposition. On the other hand, voltage imbalance caused by uneven distribution of loads over three phases is more difficult to mitigate. Large single-phase loads such as single-phase induction motors, traction systems, induction heating, etc., are typical examples that cause voltage imbalance in industrial electric power systems. In low voltage residential and/or commercial systems, single-phase loads account for the majority of power consumption. Wherever possible, efforts are made to distribute the single-phase loads uniformly over three phases. From a statistical point of view, however, distributing single-phase loads uniformly over the three phases only ensures that the expected values of the loads in each phase will be approximately equal. But it is unlikely that at a given instant, three-phase loads will be balanced because they vary in a random manner. In other words, even if the average loads in the three phases are kept the same, the instantaneous power demands in the three phases differ from each other, leading to imbalanced voltages at the point of common coupling. The degree of imbalance of a three-phase voltage is often measured based on the ratio of its negative- to positive-sequence component. This ratio is termed the unbalance factor (UBF). By assuming that the magnitudes of three-phase voltage phasors are Gaussian random variables, and that no phase angle deviations exist (i.e., 10º out of phase with each other), Pierrat and Morrison (1995) developed a probabilistic model of the voltage UBF and showed that the UBF would have a Gaussian distribution if the unbalance were mainly caused by asymmetry in system impedances, and have a Rayleigh distribution if the system impedances were symmetric. This model, though capable of providing physical insight into the causes of voltage unbalance, is somewhat restricted by its assumptions. Instead of assuming normally distributed voltage magnitudes, it seems more plausible to assume that the loads (including active and reactive powers) in the three phases will approach a jointly normal distribution as long as the number of electric appliances supplied by the power system is large enough (which is usually the case). This assumption has a theoretical basis supported by the central limit theorem, hence allowing the restrictions of the model 16

2 Random Variation of Voltage Unbalance proposed by Pierrat and Morrison (1995) to be relaxed and more accurate results to be obtained. Another advantage of using a jointly normal distribution model lies in its ability to handle correlation between all Gaussian random variables. This ability is especially important in modeling three-phase active and reactive powers since they are usually strongly correlated. This paper combines numerical techniques of load flow and Monte Carlo simulation and proposes a novel method of simulating voltage unbalance caused by fluctuation of loads. The simulation begins with generation of six correlated Gaussian random variables representing three-phase active and reactive powers. The Cholesky decomposition method is used to accomplish this task. This is followed by a load flow calculation used to determine the three-phase voltage phasors at the point of common coupling. Both the magnitude and the angle of UBF can then be obtained by means of a symmetrical component conversion. The probability density functions of UBF under several situations will be studied. II. Formulation 1. Generation of Correlated Gaussian Random Variables Let P a, P b and P c be the total active powers, and let Q a, Q b and Q c the total reactive powers absorbed by a cluster of loads in phases a, b and c of an electric power distribution system. For a sufficiently large number of loads, P a, P b, P c, Q a, Q b and Q c have approximately a jointly normal distribution. If the distribution system supplies only threephase loads, then the random variables P a, P b, P c, Q a, Q b and Q c will be strongly correlated, owing to the symmetry of variation in each phase. On the contrary, if all loads are (statistically) independent single-phase loads, then the six random variables will be uncorrelated. A realistic distribution system supplies a mixture of three-phase and singlephase loads. Hence, a real situation may be somewhere between the two extreme cases just mentioned. In order to simulate random variations of active and reactive powers in the three phases, correlated Gaussian random numbers must be generated. The following illustrates the procedure used to generate correlated Gaussian variables. Let X 1 and X be a pair of independent random variables, both of which are uniformly distributed over the interval [0,1]. Then two independent standard normal random variables (i.e., each with zero mean and unit variance) Y 1 and Y can be given by (Box and Muller, 1958) Y = ln( X ) cos( πx ), 1 1 Y = ln( X ) sin( πx ). 1 (1) () A linear transformation of Y 1 and Y, W1 Y1 = T, W Y produces two new Gaussian random variables W 1 and W with their covariance matrix c w given by σw1 ρwσw σ 1 w c w = ρwσw σw σw 1 where σ w1 and σ w are their respective standard deviations, and ρ w their correlation coefficient. It is emphasized that the covariance matrix c w is related to the covariance matrix of Y 1 and Y, which is a unit matrix, by (Peebles, 1993) t t cw = T 1 0 T = TT. 0 1 It is noted that if T is a lower triangular matrix of the form T = T T then the elements of T can be determined to be T 11 = σ, 11 w1 0, T 1 T1 = ρwσ w, T = w 1 w σ ρ. Hence, W 1 and W become W = T Y =σ Y, w w w 1 w w W = T Y + T Y = ρ σ Y + σ 1 ρ Y. (3) (4) (5) (6) (7) (8) (9) (10) If W 1 and W have means w 1 and w, respectively, then it suffices to add them to the right sides of Eqs. (9) and (10): W = W +σ Y, 1 1 w1 1, ( ) w w 1 w w ( ) W = W + ρ σ Y + σ 1 ρ Y. (11) (1) Equations (1), (), (11) and (1) enable us to generate two correlated Gaussian random variables W 1 and W with their respective means w 1 and w, standard deviations σ w1 and σ w and correlation coefficient ρ w, using the uniform random numbers X 1 and X. 17

3 Y.J. Wang The foregoing development can easily be extended to the generation of six correlated Gaussian random variables representing active and reactive powers in the three phases. Let c w represent the covariance matrix of the sixdimensional normal distribution. The covariance matrix c w can be factored into a lower triangular matrix T and an upper triangular matrix T t that is the transpose of the former. This special factorization is called Cholesky decomposition and can be written as W = [ W, W, K, ], 1 W6 t (18) Equations (10)-(18) permit generating six arbitrary correlated Gaussian random variables which represent the active and reactive powers in the three phases. The computer program for Cholesky decomposition of the covariance matrix c w is available in most FORTRAN libraries (e.g., Press et al. (199)), which largely reduces the effort involved in coding Monte Carlo simulation programs. c w = [ c ] = T T, ij 6 6 t (13). Network Reduction and Load Flow Study where c ij σwj, if i = j, = ρσ ij wiσwj, if i j T T1 T T 31 T3 T T =. T T T T T51 T5 T53 T54 T55 0 T61 T6 T63 T64 T65 T66 (14) The elements of T can be calculated using the Cholesky method by factoring matrix c w. Equations (1) and () allow six independent standard normal variates Y 1, Y,..., Y 6 to be generated from six independent uniform random variables X 1, X,..., X 6. Therefore, the correlated Gaussian random variables can be given by W = TY+ W, (15) where W is a vector composed of six correlated Gaussian random variables, Y a vector containing six independent standard normal random variables, and W a vector composed of the means of the six correlated Gaussian random variables, namely, W = [ W, W, K, ], 1 W6 t Y = [ Y, Y, K, ], 1 Y6 t (16) (17) The Monte Carlo method is a powerful tool for the simulation of randomly varying loads, and it can be combined with the conventional deterministic load flow calculation to ascertain random variation of voltages at the load buses. To do so, the deterministic three-phase load flow calculation is repeated a sufficiently large number of times. Each time, the nodal active and reactive powers are changed by generation and transformation of random numbers, as mentioned in the previous section of this chapter. The probability distributions of the nodal voltages, and thus that of the voltage UBF, can then be obtained. However, the amount of calculation would be prohibitive for any practical distribution system if repetitive three-phase load flow calculations were to be carried out by considering random variations of active and reactive powers at all the load buses. In fact, it is often the case that the voltage unbalance problem occurs only at some particular substations or feeders that supply large single-phase loads (e.g., railway traction locomotives) or unbalanced three-phase loads (e.g., arc furnaces). In this case, one may wish to confine his/her study of voltage unbalance to those buses to which unbalanced loads are connected. The notion of network reduction helps one reach this goal by representing a large network with its Norton or Thevenin equivalence while retaining buses of special interest. Network reduction is important when Monte Carlo simulation and three-phase load flow study are to be combined together to calculate the probability distribution of the UBF at a particular load bus since both methods are computationally intensive. For a (k + m)-bus system in which k buses are to be eliminated and m buses retained, its nodal equations in matrix format can be written as (19) 18

4 Random Variation of Voltage Unbalance Note that in Eq. (19), the buses to be retained are numbered as the last m nodes. Equation (19) is partitioned as shown above to obtain where S 1, S,..., S m refer to the complex powers drawn at each corresponding bus, and the asterisk (*) the conjugate. The voltage vector can be given by Eliminating V E yields Equation (1) can be rewritten as where Y Y EE P T Y V Y Y P PP 0 eq p eq V V = I ( ), IE =. IP E P PP P T 1 EE P P P P T 1 EE E ( Y Y Y Y ) V = I Y Y I. (0) (1) () T ( 0) V = [ V V L V m ] = V Z I. 1 (8) Equations (7) and (8) lend themselves to the iterative method of finding the solutions of V for known complex powers S 1, S,..., S m. In this paper, the Gauss-Seidel iterative method (Glover and Sarma, 1994) is used to determine the bus voltages. 3. Voltage Unbalance Factor The complex voltage unbalance factor is defined as the ratio of the negative-sequence to the positive-sequence component eq eq eq PP P T 1 EE P Y = Y Y Y Y, (3) + UBF = V V = τ exp( jθ), (9) ( 0) eq P P T 1 EE E I = I Y Y I. (4) It is noted that in Eqs. (3) and (4), Y eq represents the Norton equivalent admittance matrix and I (0) eq the Norton current vector. The corresponding Thévenin impedance matrix and voltage source vector can be given by where j = 1, and V and V + refer to negative-sequence and positive-sequence voltages, and τ and θ to the magnitude and the angle of the complex UBF, respectively. V and V + are given using Fortescue transformation by V = ( V + a V + av )/ 3, AT BT CT (30) Z eq = Y 1, eq (5) V + = ( V + av + a V )/ 3, AT BT CT (31) eq ( 0 ) 1 eq eq ( 0 = ). V Y I (6) An m-port Thévenin equivalent circuit is shown in Fig. 1, in which m buses are retained for study of the voltage unbalance problem. In the figure, the currents I 1, I,..., I m can be written in matrix form as I I I L I T m S1 V 1 S V T m m S V = [ ] = L 1 *,(7) where a = exp(jπ/3), and V AT, V BT and V CT refer to the three-phase voltage phasors. III. Numerical Examples Two examples will be given for the purpose of illustration. The first example is a 6.-km-long 11 kv threephase four-wire overhead feeder which is used to provide electricity from a substation to a load center. Various hypothetical conditions have been deliberately assumed in the first example in order to show the influence of the (statistical) correlation and the standard deviation of threephase active and reactive powers upon the probability distribution of the UBF at the load end of the feeder. The second example is taken from the Taiwan Power company s Tou-Liu district distribution substation, which is a 69/11 kv substation. The second example studies the probability distribution of the voltage UBF at the secondary terminals of a distribution transformer using the field recorded three-phase power data. 1. An 11 kv Three-phase Overhead Feeder Fig. 1. An m-port Thévenin equivalent circuit of the distribution system. A three-phase four-wire distribution feeder is taken as the first example to study the random variation of three- 19

5 Y.J. Wang phase voltage caused by fluctuation of loads in the three phases. The circuit model of the system is shown in Fig. 1, in which the impedance matrix Z bus has been calculated according to the arrangement of conductors shown in Fig.. The method described in Subsection II.1 was used to carry out the simulation. We simulated nine representative cases, the parameters of which are listed in Table 1. In these cases, the correlation coefficients between any two Gaussian random variables are the same and are equal to ρ. The means and the standard deviations of the active and reactive powers in the three phases are all equal to µ and σ, respectively. In Table 1, µ and σ are expressed in per unit (p.u.). The corresponding base values of the voltage and kilovoltamperes are 11.4 kv (line-to-line) and 5,000 kva (three-phase), respectively. An important point to be noted is that in all nine cases, stationary balanced power consumption is assumed; i.e., the means of the three-phase complex powers are equal. It is noted that although the assumptions for the illustrative cases have been largely simplified, the proposed simulation method is general and is applicable to a wide range of problems. A. Influence of Correlation Coefficient The nine cases listed in Table 1 can be categorized into three groups according to the degree of correlation: highly correlated (the first three, ρ = 0.9), moderately correlated (the subsequent three, ρ = 0.5) and uncorrelated (independent) cases (the last three, ρ = 0). We will compare cases 1, 4 and 7, where the means and standard deviations of the active and reactive powers are, respectively, 0.13 p.u. and p.u. for each phase while the correlation coefficients ρ are, respectively, 0.9, 0.5 and 0. The corresponding probability density functions (PDFs) of the magnitude of voltage UBF(τ) are depicted in Fig. 3. It can be seen that both the mean and the standard deviation of τ increase with decreasing ρ. In other words, the more the power fluctuations in each phase are correlated, the more likely the three-phase voltage will be balanced. To illustrate this result more clearly, variations of the mean and the standard deviation of τ versus the correlation coefficient are shown in Fig. 4. A high degree of correlation of power fluctuations in the three phases implies that a majority of the three-phase loads are supplied by the feeder, thus causing the threephase voltage to be less unbalanced. Conversely, a low degree of correlation implies that the feeder supplies a large number of independent single-phase loads. Wide fluctuation of power among the three phases can be experienced, and more severe voltage unbalance may be caused. The proposed simulation method is also capable of providing useful information on random variation of the angle (θ) of the complex UBF by which V leads V +. The statistical distribution of θ, though relatively less frequently discussed in the literature, plays an important role in some studies (e.g., Wang and Pierrat (1993)). The PDFs of θ for ρ = 0.9, ρ = 0.5 and ρ = 0 are shown in Fig. 5, where an increase in phase diversity with decreasing ρ can be observed. Fig.. Arrangement of three-phase conductors for numerical example number one. Table 1. Parameters of Nine Simulation Cases case parameters 1 ρ = 0.9, µ = 0.13 p.u., σ= p.u. same as case 1 except for σ= p.u. 3 same as case 1 except for σ= p.u. 4 ρ = 0.5, µ = 0.13 p.u., σ= p.u. 5 same as case 4 except for σ= p.u. 6 same as case 4 except for σ= p.u. 7 ρ = 0, µ = 0.13 p.u., σ= p.u. 8 same as case 7 except for σ= p.u. 9 same as case 7 except for σ= p.u. Fig. 3. PDFs of the magnitude of voltage UBF(τ) for ρ = 0.9, ρ = 0.5 and ρ = 0. 0

6 Random Variation of Voltage Unbalance. A 69/11 kv Distribution Substation Fig. 4. Variations of the mean and standard deviation of τ versus ρ, assuming σ= p.u. and µ = 0.13 p.u. for the active and reactive power of each phase. B. Influence of the Variability of P and Q The variability of power is another factor that influences voltage unbalance. The variability of a random variable is measured based on its standard deviation. When other system conditions are kept the same, power systems with lower variability of power consumption are likely to have a lower degree of voltage unbalance. Taking cases 4, 5, and 6 as examples for comparison, we notice that they have the same conditions ρ = 0.5 and µ = 0.13 p.u. while their standard deviations are p.u., p.u. and p.u., respectively. It is seen in Fig. 6 that as σ increases, both the mean and standard deviation of τ go up. Figure 7 shows the variations of the mean and the standard deviation of τ versus the standard deviation of power (σ). The PDFs of the angle (θ) of UBF shown in Fig. 8 reveal that the phase diversity of UBF increases with σ. This result is comparable with that shown in Fig. 5. In both cases, the variability and phase diversity of UBF are closely related. In the previous example, the probability properties (i.e., the means, variances and correlation coefficients.) of the three-phase active and reactive powers were deliberately assumed so that the influence of a specific parameter on the probability distribution of the UBF could be found. In this section, a real-world case will be studied. Figure 9 shows the one-line diagram of a 69/11 kv substation containing three distribution transformers. Only the transformer #3 part is shown in the figure. The low-tension side of the transformer is connected to five feeders supplying electricity to several load centers in Tou-Liu city. The three-phase active and reactive powers flowing through the transformer were recorded using a digital power analyzer. The measurement was carried out late in the autumn of 1998 over several days. Because of the memory capacity limitation of the power analyzer, an appropriate sampling period must be selected so that successive data Fig. 6. PDFs of τ for σ= 0.005, and 0.035, where ρ is 0.5 and µ is 0.13 p.u. for the three cases. Fig. 5. PDFs of the phase angle of UBF(θ) for ρ = 0.9, ρ = 0.5 and ρ = 0. Fig. 7. Mean and standard deviation of τ versus σ. 1

7 Y.J. Wang B. Monte Carlo Simulation To simulate random variation of the UBF shown in Fig., we need to model the active powers P a, P b and P c and the reactive powers Q a, Q b and Q c using correlated Gaussian random variables. It is noted that the power is neither purely deterministic as a function of time, nor is it purely probabilistic, but rather is a combination of both. The power can be resolved into a deterministic component and a probabilistic component. The former accounts for the average (expected) daily power curve, and the later for the random deviation from the expected quantity. Therefore, we can write Fig. 8. PDFs of θ for σ= 0.005, and Pt () = Pt () +ξ, i i i Q () t = Q () t +δ, i i i (3) (4) recording can be carried out without interruption. Several sessions of measurement were conducted. Each session lasted either 7 hours with a sampling period of 30 seconds, or 7 days with a sampling period of 5 minutes. The data to be studied using the probabilistic method proposed in this paper were recorded during one of these 7-hour sessions beginning at 11:00 in the morning and ending at 13:40 the next afternoon. where i = a, b or c refers to the phase specified, P i (t) and Q i (t) to the deterministic components of active and reac- A. Voltage UBF Calculated based on Recorded Active and Reactive Power Data Figure 10 shows the recorded active powers P a, P b and P c, and reactive powers Q a, Q b and Q c. All the power data are expressed as per-units of the megavoltamperes per phase. The base value of the megavoltamperes per phase equals 5 MVA in this example, being one third of the rated three-phase megavoltamperes of the transformer. It is noted that Fig. 10(b) shows a sudden increase and a sudden decrease in reactive power at about the 1,400th and,600th time steps, respectively. The sudden changes in reactive power were caused by switching of capacitor banks in the substation. Figure 11 shows the transformer low-tension side line-to-line voltages V ab, V bc and V ca in per-units (on the line-to-neutral voltage base) which were obtained using a three-phase load flow program in Fortran. In the load flow program, the 69 kv bus was taken as the slack bus, and the power data shown in Fig. 10 as power injected at the load buses. The star-delta (Y- ) three-phase model of the transformer proposed by Arrillaga et al. (1983) was used in the program, and the iterative method described by Eqs. (7) and (8) was coded to find the voltages at the 11 kv buses. The corresponding voltage UBF was calculated using Eqs. (0)-() and is depicted in Fig. 1(a). Fig. 9. One-line diagram showing a distribution transformer connected to 69 kv and 11 kv buses.

8 Random Variation of Voltage Unbalance (a) Fig. 11. Evolution of the three-phase line-to-line voltages at the low voltage side of the transformer. (b) Fig. 10. Recorded three-phase power carried by the transformer: (a) three-phase active power, (b) three-phase reactive power. tive powers, and ξ i and δ i to their probabilistic components. The deterministic components P i (t) and Q i (t) can be obtained by using a low-pass filter to filter out the random variations of P i (t) and Q i (t), and the probabilistic components ξ i and δ i can be modeled by Gaussian random variables as described in detail in Section II. Equations (1)- (18) then allow ξ a, δ a, ξ b, δ b, ξ c and δ c to be generated using a random number generator. The foregoing modeling of P i (t) and Q i (t) is indeed an application of random processes. P i (t) and Q i (t) are normal random processes, and ξ a, δ a, ξ b, δ b, ξ c and δ c are strict-sense stationary random processes. The active and reactive powers P a, P b, P c, Q a, Q b and Q c simulated according to Eqs. (3) and (4) were taken as the inputs to the three-phase load flow calculation program to compute the low-tension side voltages of the distribution transformer. And the corresponding voltage UBF was also computed and is depicted in Fig. 1(b). Comparison of Fig. 1(a) and Fig. 1(b) reveals that the proposed method is capable of closely simulating both deterministic and probabilistic variation of the unbalance in the three-phase voltage. While Fig. 1 provides the reader with a qualitative evaluation of the proposed method, Fig. 13, which compares the PDFs and the CDFs of the UBF obtained from measured and from simulated power data, may offer a quantitative validation of the method. IV. Conclusions This paper has proposed a method for simulating three-phase voltage unbalance caused by random fluctuation of loads in the three phases. Based on the central limit theorem, the active and reactive powers drawn in the three phases from a distribution feeder have been modeled by six correlated Gaussian random variables. This paper has explained how to generate the correlated Gaussian random variables using the Cholesky decomposition method and how to use the iterative method to solve the voltage phasors at the load buses. Unbalance of the three-phase voltage can then be evaluated. The complex voltage UBF has been adopted as an index of the degree of voltage unbalance. Probability density functions of the magnitude (τ) and the angle (θ) of UBF can be easily obtained using the proposed method. The influence of the correlation of active and reactive powers in the three phases has been studied. It has been found that the mean of UBF decreases as the correlation increases. Also studied was the influence of the variability of the loads on UBF. For highly fluctuating power 3

9 Y.J. Wang (a) (b) (a) Fig. 13. Comparison of (a) the PDFs and, (b) the CDFs of UBF, calculated using field recorded data (triangular dots) and using the Monte Carlo simulation method (solid lines). (b) Acknowledgment The author is grateful to the National Science Council, R.O.C., for research grant NSC E Fig. 1. (a) Evolution of UBF obtained using the voltage data shown in Fig. 11; (b) evolution of UBF obtained using the Monte Carlo simulation method. consumption in the three phases, it is likely that a high degree of voltage unbalance will exist. The PDF of θ provides important information about the phase diversity of UBF. It has been found that for voltage unbalance caused by load fluctuation, the variability and phase diversity of UBF are closely related. In addition to a fundamental example based on an 11 kv three-phase overhead feeder, field recorded threephase power data from a 69/11 kv distribution substation have also been used as a more advanced example to illustrate how the proposed Monte Carlo simulation technique can be incorporated into the load flow calculations to take into account both the deterministic and the probabilistic variations of the loads and those of the resulting voltage unbalance. Good agreement between measured and simulated results has been obtained. References Arrillaga, J., C. P. Arnold, and B. J. Harker (1983) Computer Modelling of Electrical Power Systems. Wiley, Chichester, NY, U.S.A. Box, G. E. P. and M. E. Muller (1958) A note on the generation of random normal deviates. The Annals of Mathematics Statistics, 9, Glover, J. D. and M. Sarma (1994) Power System Analysis and Design, nd Ed., pp PWS Publ. Co., Boston, MA, U.S.A. Peebles, P. Z., Jr. (1993) Probability, Random Variables, and Random Signal Principles, 3rd Ed. McGraw-Hill, New York, NY, U.S.A. Pierrat, L. and R. E. Morrison (1995) Probabilistic modeling of voltage asymmetry. IEEE Transactions on Power Delivery, 10, Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery (199) Numerical Recipes in FORTRAN, nd Ed., pp Cambridge University Press, New York, NY, U.S.A. Wang, Y. J. and L. Pierrat (1993) Probabilistic modelling of current harmonics produced by an ac/dc converters under voltage unbalance. IEEE Trans. on Power Delivery, 8, Wang, Y. J., L. Pierrat, and R. Feuillet (199) An analytical method for predicting current harmonics produced by an ac/dc converter under unbalanced supply voltage. European Transactions on Electrical Power Engineering,,

10 Random Variation of Voltage Unbalance Cholesky decomposition 5