Three Phase Power Flow for Distribution Systems ... - Semantic Scholar

14th PSCC, Sevilla, 24-28 June 2002

Session 11, Paper 3, Page 1

Three Phase Power Flow for Distribution Systems with Dispersed Generation A. Abur Department of Electrical Engineering Texas A&M University College Station, TX 77843, U.S.A. [email protected]

Abstract - This paper describes the development of a three phase distribution system power flow program. The program is developed without any radial topology assumptions and thus can be used to solve both radial as well as meshed networks without any restrictions. Distributed generation, any combination of different types of loads including constant power, constant current and constant impedance with floating or grounded neutral can be modelled at a given bus. Voltage regulators are also modelled by three phase tap changing transformers, where taps on individual phases can be adjusted independently to satisfy the desired voltage correction along the feeder. The program can handle unbalanced loads and generation as well as un-transposed lines and missing phases. Modelling details for all system components, computational issues involved in factorizing the Jacobian having a special structure and in adjusting the control variables are described. Different size systems ranging from 14 to 155 buses are used to demonstrate the performance of the program under different operating conditions involving unbalanced loads, voltage regulator settings, different combinations of load types, remote generation and different network topologies such as meshed versus radial systems.

Keywords - Power flow, unbalanced operation, three phase modelling, distribution systems, dispersed generation. 1 INTRODUCTION

T

HREE phase power flow solutions of distribution systems are needed both for planning and operation studies. Power flow studies for balanced operation can be carried out using very efficient methods [1, 2]. Three phase power flow is commonly considered exclusively for the distribution systems, where single or double phase circuits may be present and loads may not be always balanced between the three phases. Moreover, distribution systems used to be commonly configured as radial systems, containing very few or no loops and a single power source at the beginning of the main feeder. Hence, a number of solution schemes have been proposed to take advantage of this special structure and primarily unidirectional power flow along the radial tree structure of the feeders [3, 4, 5]. Recent increase in the number of independent power generators supplying power to the system at distributed and remote locations, necessitates three phase power flow studies for more general systems. These systems may not be strictly radial and their feeders may carry power flows whose direction may change as a function of loading and

H. Singh, H. Liu and W.N. Klingensmith Thomas A. Edison Technical Center Cooper Power Systems Franksville, WI 53126-0100, U.S.A. hsingh, hliu,[email protected]

remote generation. Furthermore, operation of such systems will require closer monitoring and control. Control functions such as voltage regulation via tap changing transformers or voltage regulators, and reactive power adjustment at the remote generator terminal buses will have to be accounted for. As the distribution automation becomes more wide spread, software tools that will aid these automation functions will be needed. This need is addressed by various studies so far. In [6], a detailed formulation and modelling for the three phase power flow problem is given and a fast decoupled version of the Newton’s method is presented for its solution. A Zbus based method that models generators as constant PQ devices and has a single specified voltage at the substation bus, is developed in [7]. The compensation method introduced in [3] is extended to the three phase systems in [8]. This is well suited for strictly radial topologies, but slows down for increasing number of loops in the network. A Newton based method and branch equations in rectangular coordinates are used to formulate and solve the three phase harmonic power flow in [9]. This approach is further developed and incorporated into EMTP to correctly initialize the three phase transient simulations [10]. More recently, a three phase current injections method is shown to have good convergence properties [11] for cases that do not involve regulators. This paper reports on the development of a three phase power flow program which addresses the above mentioned concerns of general power system topologies. Various modelling and computational techniques that are previously developed and successfully employed in single phase as well as three phase power flow solutions are utilized in this development. The paper is organized such that the requirement specifications for a three phase power flow are first introduced. These relate both to the power system modelling as well as the operational constraints that need to be accounted for during the solution. The main solution algorithm and details of the implementation of the various controls are presented next. Simulation results followed by the conclusions will end the paper. 2 Specifications The specifications of a three phase power flow program may be stated under two main categories, modelling and implementation. Modelling specifications provide the description of the type and operating characteristics of the power system components that should be recognized and properly modelled by the program. Implementation specifications indicate the features that allow the user to obtain



solutions restricted by various operating limits. 2.1

Modelling 1. Loads: Single, double and three phase loads could exist at any bus. Three types of loads can be specified and any combination of them may exist at each bus: Constant Power: Real and reactive power injections at the bus are specified for each phase. Constant Current: Current magnitude and its power factor are specified for individual phases at the bus. At each iteration, these values are converted into equivalent real and reactive power injections using the most recently calculated value of the bus voltage. Upon convergence, the specified current and its associated power factor will be satisfied. Constant Impedance: These types of loads may be connected in delta (∆), Y with floating neutral, Y with grounded neutral or Y with a neutral grounded through an impedance, as shown in Figure 1. ∆ connection is equivalent to Y with floating neutral and hence treated within the model of Figure 1. Admittances of the phase i load and the neutral-ground connection branch are labelled by yi and y0 respectively in the figure. They are modelled using a (3x3) admittance matrix Yload , which is added to the block diagonal element of the three phase bus admittance matrix of the system. If the neutral is connected to the ground through a non zero impedance or it is disconnected (y0 = 0), then Yload elements will be given by: Yload,ii

=

Yload,ij

=

where : ∆

=

yi2 i = 1, 2, 3 ∆ yi · yj i 6= j − ∆ 3 X y0 + yi yi −

i=1

and, if the neutral is solidly grounded: Yload,ii Yload,ij

= =

yi i = 1, 2, 3 0 i 6= j

Phase-A

Phase-B

Phase-C

y1

y2

y3

y0

Figure 1: Constant Impedance Type Load

2. Generation: Remote generation at any of the system buses should be modelled as a controlled voltage source whose total real power output is specified. Generator terminal voltage is typically controlled via the specification of the positive sequence component only. This is accomplished via the use of a three phase synchronous generator model with balanced excitation voltage [6]. Consider a generator i, connected to the system at the three phase bus (j, k, `) as shown in Figure 2. In the steady state, the generator is modelled by a balanced three phase excitation voltage source behind its three phase synchronous impedance matrix Zs . This matrix can be specified either in the phase domain or more commonly in the sequence domain. Note that the posi itive sequence component (Vpos of Eq.(1) ) of the terminal phase voltages j, k, ` is to be held constant along with the total three phase real power output (Pispec of Eq.(2) ) of the generator. Since the internal three phase excitation voltage source has balanced voltages, only phase-A voltage Eia (magnitude and phase) is included in the unknown variable vector. i Vpos

=

Pispec

=

α Ii Vj,k,`

1 (Vj + αVk + α2 V` ) 3 Real{Vj,k,` · Ii∗ }

(1) (2)

j 2π/3

= e = [Iia Iib Iic ] = [Vj Vk V` ] I ia

j

E ia Generator Balanced Internal Voltages

I ib E ib

E ic

Zs I ic

k

Generator Terminal Nodes

l

Figure 2: Distributed Generator Model

3. Transformers: Transformers with fixed or controllable taps have to be modelled. They may be three phase units or may be single phase units connected in any number of the individual phases depending on the feeder configuration. Assuming that they operate in their linear region and neglecting their excitation currents, transformers can be modelled by the nodal admittance matrices [12, 6]. Consider a transformer connected between buses k and m, and whose admittance matrix is given as Ytr . If the transformer is tapped and bus k is the tap side, then the nodal admittance matrix for the two bus (k, m) subsystem will be given by: · −1 ¸ k A Ytr A−1 −A−1 Ytr Ynode = m −Ytr A−1 Ytr where, A is the (3x3) diagonal matrix whose diagonal entries are the off-nominal taps for each phase.



The details of obtaining Ytr for different three phase transformer connections can be found in [6]. This model is used for representing both the fixed tap transformers as well as controlled tap voltage regulators. Taps can be adjusted to maintain the voltage at the regulator terminals or at any other bus downstream along the feeder. 4. Shunt Capacitors/Reactors: Single or three phase shunt elements directly connected between a bus and the ground should be modelled. Three phase units can be connected in Y (grounded or with floating neutral) or delta. They are modelled in the same way as the constant impedance type loads described above. 2.2

Implementation of Constraints 1. Tap/Voltage Limits: Voltage regulators are represented as variable tap transformers with controllable terminal voltages. This requires the adjustment of taps during the iterative solution of the power flow equations, so that the specified bus voltages are maintained. In the current implementation, this is accomplished by using the approximate sensitivities of the taps to the bus voltages. However, this feature can be easily modified to maintain any function of the system bus voltages, for instance the difference between the secondary and primary, if needed. Details of automatic tap adjustments will be presented in the next section. 2. Q-limits: Remote generators are modelled as three phase sources with specified total real power output and positive sequence terminal voltage. Reactive power output Q, of the generator is adjusted to maintain the specified voltage while respecting the limits on Q. At each iteration, these limits are checked and if exceeded, the bus type is switched back to PQ where the exceeded limit is used as the specified Q for this PQ bus. 3 Solution Algorithm

Three phase power flow problem is solved using the Newton-Raphson iterative solution method. Consider a system of (N+1) buses, Ng of which are generator buses and one slack bus. Using the generator model of Figure 2, each generator terminal bus is connected to an internal generator bus. Thus, the generator terminal buses will become of PQ type with net zero PQ injections. On the other hand, the internal generator buses are constrained to have balanced excitation voltages and therefore only phase-A node voltage will have to be maintained as an unknown. Hence, the solution should be obtained for the 3N and Ng voltage phasors for the PQ type nodes and generator internal nodes respectively. The power flow problem can then be formulated using four sets of equations. These are: ∆PL

=

P sp − P cal (V, θ) = 0

(3)

∆PG

=

PGsp − PGcal (V, θ) = 0

(4)

∆QL

=

sp

Q

−Q

cal

(V, θ) = 0

(5)

∆VG

=

VGpos − f (V, θ) = 0

(6)

where: T T [θT V T ]T = [θL θG VLT VGT ] T θL = [θ1 · · · θ3N ], T θG = [θ3N +1 · · · θ3N +Ng ], VLT = [V1 · · · V3N ], VGT = [V3N +1 · · · V3N +Ng ], Pisp , Pical (V, θ) are the specified and calculated real power injection at node i, cal Qsp i , Qi (V, θ) are the specified and calculated reactive power injection at node i, sp cal (V, θ) are the specified and calculated total real PGi , PGi power output of generator i, pos VGi , fi (V, θ) are the specified and calculated value of the positive sequence voltage magnitude at generator i. While it is possible to apply the well documented decoupling assumptions and obtain a fast decoupled version of the above formulation [2, 6], the full version is maintained in order to avoid possibility of numerical instability due to the special characteristics of the distribution systems. One common feature of the distribution or subtransmission systems is the existence of high R/X ratio branches which violates the assumptions used in decoupled formulation. The mismatch equations given by Eq.(3-6) can be expressed in compact form as a nonlinear vector equation given below: w(X) = 0 (7) where, X T = [θT V T ] wT (X) = [∆PLT ∆PGT ∆QTL ∆VGT ]. Newton-Raphson method can then be used to iteratively solve Eq.(7) as follows: ∆W = −w(X k ) = [J] · [∆X]

(8)

where, k [J] = ∂w(X) ∂X , Jacobian matrix evaluated at X , k X = Solution vector at iteration k, ∆X = X k+1 − X k . 3.1 Structure of the Jacobian Based on the mismatch equations, the Jacobian can be formed as follows:

J=

∆PL ∆PG ∆QL ∆VG

∆θL ∆θG JPL θL JPL θG  JPG θL JPG θG   JQL θL JQL θG JVG θL 0 

∆VL ∆VG JPL VL JPL VG JPG VL JPG VG JQL VL JQL VG JVG VL 0

   

Note that, the sub-matrices of the block row ∆PG are very sparse, since the internal generator buses have only three connections to the rest of the network, namely the three phase nodes of the associated generators. The way the Jacobian is ordered, there are zero pivots at the bottom right corner, however these will be filled during the triangular factorization of the Jacobian.


3.2


0

Initialization

The solution can be initialized from the 3-phase balanced flat start conditions, i.e. all node voltages are assigned 1.0 pu magnitude and 0, -120, 120 degree phase angles for the a,b and c phases respectively. Slack bus voltage as well as all internal generator bus voltages are maintained balanced throughout the solution. Constant current loads are converted into equivalent power injections evaluated at the assumed flat start voltages. Their values are updated at each subsequent iteration as the voltage solution is changed. Controlled taps of the voltage regulators are initially set equal to their nominal values, which is usually equal to 1.0. 3.3

Sparsity Methods and Ordering

100

200

300

400

500

600

700

800 0

Solution of Eq.(8) involves sparse triangular factorization of the Jacobian, followed by the forward and back substitutions. The structure of J is not symmetric unlike the case of the single phase power flow problem. Furthermore, zero pivots appearing at the bottom right corner do not allow unrestricted pivoting just based on sparsity considerations. In order to exploit sparsity and still avoid zero pivots, a modified Tinney-2 [13] ordering is applied to the Jacobian matrix. The ordering scheme chooses the next pivot based on the nonzero count of the rows disregarding the fact that the matrix may not be symmetric. Each time a generator terminal node is chosen as the pivot, the associated internal generator node is ordered as the immediate next pivot. This way, all zero pivots are guaranteed to be eliminated after the elimination of the pivot associated with the generator terminal bus and the sparsity is still preserved. This strategy is found to yield the best results in terms of the computational and storage efficiency. As an illustration, consider the IEEE 118 bus system, modelled as a 3-phase network. The corresponding unordered Jacobian sparsity structure is shown in Figure 3. Note that the bottom right corner has a (43x43) zero submatrix corresponding to the 43 PV-nodes in this system. 0

100

200

300

400

500

600

700

800

Figure 4: 118-Bus Test System Jacobian Ordered with Modified Tinney2 Ordering 0

100

200

300

400

500

600

700

800 0

100

200

300

400

500

600

700

800

Figure 5: 118-Bus Test System Jacobian After Factorization

Figure 4 shows the structure of J after the above described ordering. The Table-of-Factors of the ordered J, obtained after its factorization is given in Figure 5. This ordering strategy effectively maintains the sparsity yet avoids pivoting during factorization. 3.4 Solution Adjustments

100

1. Voltage Regulators:

200

Three phase transformers with variable taps, are used to represent the voltage regulators. Either the secondary voltage or any other bus voltage along the feeder can be specified as the controlled bus voltage by the regulating tap. The error feedback method that is successfully utilized in single phase power flow solutions [2, 14] is employed for tap adjustments. The amount of tap correction is calculated by: ∆tij = ∆Vk /sk

300

400

500

600

700

800 0

100

200

300

400

500

600

Figure 3: 118-Bus Test System Unordered Jacobian

700

800

where ∆tij is the required tap correction for transformer connected between i− j, sk is the sensitivity



factor and ∆Vk is the mismatch between the specified and computed value of the voltage magnitude at the controlled bus. The sensitivities vector S is approximated as in [14]:

Solve the Mismatch Equation

Update [X]

S = J −1 · y Compute [ ∆ W]

where J is the Jacobian, y T = [0 . . . − b/t . . . b/t . . . 0] b = susceptance of the transformer, t = most recent value of tap.

N

Y

The only two nonzero entries in y will be at rows 3N + Ng + i and 3N + Ng + j for a transformer connected between nodes i − j. Note that, sparse factors of J are used along with fast forward/back substitutions to find the k-th element of the sensitivity vector S, i.e. sk .

Bus Type Switch

The overall solution algorithm is summarized in the flowchart given in Figure 6. The two thresholds t1 and t2 are used to initiate the solution adjustments (Q-limits, taps) and to terminate the iterative solution respectively. Their values are chosen as t1 = 0.1 and t2 = 0.0001 for all the presented simulation results. 4 Simulation Results The three phase power flow program is evaluated using test power systems of varying sizes and characteristics. Test systems 1 through 4 are created using the commonly known IEEE single phase systems of sizes 14 through 118 [15]. The positive sequence network data are converted into equivalent three phase system data by assuming transposed lines with zero sequence parameters that are equal to three times the positive sequence values and also assuming balanced loads. Note that these systems are meshed networks and contain several generators with Q-limits and off-nominal tap transformers. In addition to these systems, two others are created mainly to represent radial distribution systems. Figure 7 shows a small 15-bus distribution system with remote generation, voltage regulator, unbalanced loads, 2-phase and 1-phase feeders mixed with 3phase feeders. A much larger distribution system having 155 buses and a configuration similar to the 15-bus example is also created.

N

Q-limit Check ? Y

N

Calculate S Adjust Taps

2. Remote Generator Q-limits: The experience with the single phase power flow solutions [2] implies that the Q-limit adjustments should be attempted after the solution is sufficiently converged. Hence, a threshold is used on the power mismatches so that premature Q-limit and bus type adjustments can be avoided. Once the violated limits on Q generation are detected, then the corresponding bus type is converted to PQ and the injected Q is fixed at the limit value. Similarly inverse procedure is used to backup previously converted buses whose voltages get corrected during subsequent iterations.

[ ∆ W] < t1 ?

Tap/Voltage Check ?

Y

N [ ∆ W] < t2 ?

Y STOP

Figure 6: Flowchart of the Three Phase Power Flow Algorithm

Table 1 presents the main characteristics of these systems and their corresponding Jacobian. The number of “Nodes” in the table refers to the total number of single phase nodes whose voltage solutions are to be found. Table 2 shows the iteration counts and solution times for all the tested systems. Also shown are the adjustments done to enforce the V limits for generators and voltage regulator (tap) controlled buses. All of the first four test systems’ solutions for the balanced loading conditions are checked against the provided power flow solutions for these systems in [15] and computational accuracy of the results are verified. Subsequently, these systems are subjected to unbalanced loads and non-symmetric lines.

Test System 1 2 3 4 5 6

Buses N +1 14 30 57 118 15 155

Gen. Ng 4 5 6 43 1 11

Table 1: Test System Characteristics

Taps 3 4 17 9 1 11

Solved Nodes 43 92 174 351 43 473

Rows of J 86 184 348 702 86 946



Unbalanced Load 1

2

3

4

5

6

7

8

9

13

14

10

t:1 Volt. Reg.

SLACK

G REMOTE GENERATOR.

11

12

2-Phase

2-Phase

1-Phase

15

1-Phase

Figure 7: Diagram for the 15-bus Distribution System

Test System 1 2 3 4 5 6

Cpu Time (seconds) 0.05 0.05 0.17 0.38 0.05 0.71

Iteration Count 4 5 3 3 7 18

Q-limits Adjusted N Y N N Y Y

Taps Adjusted N N N N Y Y

Table 2: Solution Times (seconds) and Iteration Counts

Convergence characteristics of the solution algorithm is found not to be significantly affected by unbalanced loads or structurally non-symmetric lines, such as those having single or two phases only. For instance, test system 4, which has 118 buses, is modified to create two cases: one, where lines 110-111, 110-112, 103-110, 8586, 86-87, 75-118, 118-76, 76-77 and 71-73 are replaced by single phase lines, and the other where the loads on two of the three phases of buses 6,15,27,42,62,90, and 116 are removed creating highly unbalanced loading of the overall system. The program converged in 3 iterations for both cases with no significant change in the cpu time compared to the balanced case. On the other hand, cases involving a large number of voltage and Q-limit enforcements via regulator tap changes or bus type switching, require more iterations. Case 6 in Table 2 is such a case, where several extra iterations are needed to maintain Q and V within their limits via tap and bus type adjustments. Finally, test system 5 is used to demonstrate all the control features of the program when solving a distribution system, where: • There are 3-phase, 2-phase and 1-phase feeders as shown in Figure 7, • A single phase constant current type load at bus 15, with 0.14 pu magnitude and a p.f. of 0.7, • A mixture of constant S, Z, and I type loads at bus 3, • Severely unbalanced load at bus 6, phases A and C having no real power loads, • A constant-Z type load connected in Y with a floating neutral at bus 10, • A remote generator that maintains the positive sequence voltage of bus 5 at 1.035 pu. • A voltage regulator keeping the bus 8 voltage within 1.02 and 1.03 pu limits.

The solution requires 0.06 seconds and 9 iterations during which several Q-limit and tap adjustments are made. The three phase voltage solution is listed in Table 3. Note that the remote generator supplies the required total power while the magnitude of the positive sequence voltage at its terminal bus 5 is kept at 1.035 p.u. and the bus voltage at bus 8 in all three phases are within specified limits. Tables 4 and 5 contain the calculated load current and power flows at bus 10 and the single phase bus 15. Zero sequence current of ungrounded Y load at bus 10 is zero as expected. Also, the single phase load current and its power factor at bus 15 are maintained at their specified values of 0.14 and .70 as shown in Table 5.

Bus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Va 1.050 0.984 0.968 0.995 1.044 1.061 1.067 1.021 1.050 1.073 1.045 1.049 1.050 1.051 1.049

θa 0.00 -11.62 -22.65 -30.45 -36.71 -39.08 -41.01 -41.26 -42.96 -43.93 -39.82 -42.15 -43.74 -44.45 -44.78

Vb 1.050 0.980 0.961 0.983 1.029 1.031 1.030 1.029 1.051 1.067 1.008 0.989 0.974 – –

θb -120.00 -131.53 -142.56 -150.41 -156.77 -159.66 -161.58 -161.83 -163.40 -164.27 -159.00 -160.39 -161.08 – –

Vc 1.050 0.984 0.966 0.989 1.032 1.041 1.045 1.020 1.047 1.067 1.027 – – – –

θc 120.00 108.79 98.17 90.72 84.71 82.89 81.30 81.11 79.80 79.21 83.74 – – – –

Table 3: Three Phase Voltage Solution for the 15-bus Distribution System

a Ireal b Ireal c Ireal 0.197 -0.062 -0.135

a Iimag b Iimag c Iimag 0.085 -0.137 0.052

0 Ireal + Ireal − Ireal 0.000 0.153 0.044

0 Iimag + Iimag − Iimag 0.000 0.063 0.022

Pa Pb Pc 0.089 0.104 0.028

Qa Qb Qc -0.213 -0.123 -0.152

Table 4: Calculated Load Current and Power Flow at bus 10.

14th PSCC, Sevilla, 24-28 June 2002 a Ireal -0.0009

a Iimag -0.1400

a Imag 0.14

p.f.a 0.70

Session 11, Paper 3, Page 7 Pa 0.1028

Qa 0.1049

[6] J. Arrillaga and C.P. Arnold, “Computer Modelling of Electrical Power Systems”, John Wiley & Sons, 1983.

Table 5: Calculated Load Current and Power Flow at bus 15.

5 Conclusions This paper presents a three phase power flow algorithm that is implemented for the solution of large scale power distribution systems with dispersed generation sources. The developed program can be used to solve meshed as well as radial systems, with unbalanced loads and structurally non-symmetric feeders. Automatic adjustment of voltage regulating transformer taps to maintain voltages at designated system buses while respecting tap limits is also implemented. Remote generators can be set to operate at specified terminal voltage and the required reactive power output is adjusted within allowable limits. The program is tested on different size and type of power systems with very satisfactory results. REFERENCES [1] W.F. Tinney and C.E. Hart, “Power Flow Solution by Newton’s Method”, IEEE Transactions on Power Apparatus and Systems, Vol.PAS-86, Nov. 1967, pp.1449-1460. [2] B. Stott and O. Alsac., “Fast Decoupled Load Flow”, IEEE Transactions on Power Apparatus and Systems, Vol.PAS-93, No.3, May/June 1974, pp.859869. [3] D. Shirmohammadi, H.W. Hong, A. Semlyen and G.X. Luo, “A Compensation-Based Power Flow Method for Weakly Meshed Distribution and Transmission Networks”, IEEE Transactions on Power Systems, Vol.3, No.2, May 1988, pp.753-762. [4] F. Zhang and C.S. Cheng, “A Modified Newton Method for Radial Distribution System Power Flow Analysis”, IEEE Transactions on Power Systems, Vol.12, No.1, February 1997, pp.389-397. [5] A.G. Exp´osito and E.R. Ramos, “Reliable Load Flow Technique for Radial Distribution Networks”, IEEE Transactions on Power Systems, Vol.14, No.3, August 1999, pp.1063-1069.

[7] T-H. Chen, Mo-Shing Chen, K-J. Hwang, P.Kotas and E.A. Chebli, “Distribution System Power Flow Analysis - A Rigid Approach”, IEEE Transactions on Power Delivery, Vol.6, No.3, July 1991, pp.11461152. [8] C.S. Cheng and D. Shirmohammadi, “A Three-Phase Power Flow Method for Real-Time Distribution System Analysis”, IEEE Transactions on Power Systems, Vol.10, No.2, May 1995, pp.671-679. [9] W. Xu, J.R. Marti and H.W. Dommel, “Multiphase Harmonic Load Flow Solution Technique”, IEEE Transactions on Power Systems, Vol.6, No.1, Feb. 1991, pp.174-182. [10] J.J. Allemong, R.J. Bennon and P.W. Selent, “Multiphase Power Flow Solutions Using EMTP and Newtons Method”, IEEE Transactions on Power Systems, Vol.8, No.4, Nov. 1993, pp.1455-1462. [11] P.A.N. Garcia, J.L.R. Pereira, S. Carneiro, Jr., V.M. da Costa, and N. Martins, “Three-Phase Power Flow Calculations Using the Current Injection Method” IEEE Transactions on Power Systems, Vol.15, No.2, May 2000, pp.508-514. [12] V. Brandwajn, H.W. Dommel and I.I. Dommel, “Matrix Representation of Three-Phase N-Winding Transformers for Steady-State and Transient Studies”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No.6, June 1982, pp.13691378. [13] W.F. Tinney and J.W. Walker, “Diret Solutions of Sparse Network Equations by Optimally Ordered Triangular Factorization”, Proceedings of the IEEE, Vol.55, No.11, Nov. 1967, pp.1801-1809. [14] S-K Chang and V. Brandwajn, “Adjusted Solutions in Fast Decoupled Load Flow”, IEEE Transactions on Power Systems, Vol.3, No.2, May 1988, pp.726733. [15] Power System Test Case Archive, University of Washington, Seattle, Washington, USA. (http://www.ee.washington.edu/research/pstca).