A Three-Phase Power Flow Solution Method for ... - IEEE Xplore

A Three-Phase Power Flow Solution Method for Unbalanced Distribution Networks Salem Elsaiah, Student Member, IEEE, Mohammed Benidris, Student Member, IEEE, and Joydeep Mitra, Senior Member, IEEE Electrical and Computer Engineering Department- Michigan State University East Lansing, Michigan 48824, USA email: {elsaiahs,benidris,[email protected]} Abstract— This paper introduces a methodology for solving power flow problem of both balanced and unbalanced radial distribution networks. Two matrices the line Primitive Impedance Matrix (PIM) and the Branch-Current Matrix (BCM) are developed to obtain the power flow solution for the given network. The special topological characteristics of the distribution network have been taken into consideration while developing these matrices. Due to its distinct features, the optimal node ordering which is necessary in the forwardbackward substitution method, LU-decomposition, or Y-bus formulation and factorization are no longer needed in the proposed approach. The proposed method has been tested on a balanced 33-bus network and an 8-bus unbalanced network. The results of the proposed method were found similar when compared with some other methods presented in the literature, while requiring less computational time.

I.

INTRODUCTION

Power flow solution is a crucial tool for the steady-state analysis of any electric power system. The main objective of the load flow solution is to determine the voltage phase and magnitude at each bus as well as the real and reactive powers flowing in each line for specific loading conditions. In the literature three conventional methods have been widely used to solve the power flow problem. These methods are the Gauss-Seidel (GS) iterative method, the Newton-Raphson (NR) method, and the fast-decoupled (FDLF) method [1,2]. It has been observed, however, that these conventional power flow methods, which were essentially developed for solving power flow problem at the transmission level, can encounter convergence problems when applied directly to distribution systems unless some treatments are introduced into them. The problem of convergence can be attributed in part to the special features of the electric power distribution network. Some of these prominent features include, but are not limited to, the following:  

Radial structure or nearly radial structure (weakly meshed). High ratio that may cause the NR method and the FDLF methods to diverge.



 

Untransposed or rarely transposed lines that make the ignorance of the mutual coupling between phases unacceptable for certain applications. Unbalanced loads along with single-phase and double-phase laterals. Unbalanced distributed loads.

These features combined with the large number of nodes and branches of the distribution network make such a system fall into the category of ill-conditioned power networks. These ill-conditioned systems should be analyzed on three-phase basis rather than single-phase basis. A significant amount of research has been done to develop robust and reliable distribution power flow algorithms over the past two decades [3-5]. Such algorithms are required by several real-time engineering applications that include the operational and operating stages as reported in the IEEE tutorial course on Power Distribution Planning [6]. For example, distribution feeder reconfiguration, optimum location of dispersed sources, and reactive power control and optimization need robust, flexible, and efficient power flow programs. Therefore, an acceptable power flow program should be quick enough to deal with real-time applications in which repeated power flows are required, the memory space should also be suitable, and it should attain some sort of versatility and simplicity as well. Zimmerman and Chiang presented a fast decoupled load flow method in [7]. In this approach, a set of nonlinear power mismatch equations are formulated and then solved by Newton’s method. The advantage of this method was that it ordered the laterals instead of buses; hence the problem size has been reduced to the number of system’s laterals. Use of laterals as variables instead of nodes makes this algorithm more efficient for a given system topology, yet it may add some difficulties if the network topology is changed regularly, which is the case in distribution networks as a result of the switching operation. In [8], Zhang and Cheng used Newton’s method to solve the distribution power flow problem. Even though the Jacobian matrix has to be

computed only once, shunt capacitors were not included in this method. Baran and Wu [9] developed a method for solving distribution system power flow by solving three equations representing the voltage magnitude, real power, and reactive power. In this approach, only simple algebraic equations are utilized in the developing of the Jacobian matrix and the power mismatches. The formulation of the Jacobian matrix in every iteration, however, turns to be computationally cumbersome in terms of execution time and storage requirement. Kersting and Mandive [10] and Kersting [11] suggested a methodology to solve the distribution system power flow problem based on Ladder-Network theory in the iterative routine. This approach has the advantage that it is derivativefree and uses basic circuit theory laws. However, ladder network method computes the bus voltages twice for a single iteration. Goswami and Basu presented a direct approach to obtain a solution for the distribution system power flow for both radial and meshed networks in [12]. The method was applied also for balanced and unbalanced practical networks. The main advantage of this approach is that convergence is guaranteed for any realistic distribution system. The disadvantage, however, is that no node in the system can serve as a junction for more than three network branches, which limits the use of this approach. A compensation based method is proposed in [13] to solve the balanced distribution system power flow problem based on Kirchhoff’s laws. In this approach, a simple numbering scheme is implemented to enhance the computational efforts. Then, forward sweep/backward sweep is applied to obtain the power flow solution. The same method was extended to include the weakly meshed networks. In doing so, the grid was first broken to a number of points- ―breakpoints‖- to have a simple radial network. The radial network was then solved by the direct application of Kirchhoff’s current and voltage laws. The effectiveness of this method, however, diminishes as the number of breakpoints goes up. As a result, the application of this method to the weakly meshed networks was practically restricted. In [14], Cheng and Sharimohammadi have directly extended the work done in [13] to include the case of unbalanced distribution networks. The method emphasized on the modeling of various distribution system components and was successfully applied for large systems. Teng has proposed numerous methods for solving distribution load flow [15-17]. One method [15] is based on the optimal ordering scheme and triangular factorization of the bus-admittance matrix, (Ybus). Inherently, this approach is a combination of the modified GS-method and the Z-implicit method. Basically, the approach used the Y-bus factorization, hence large computational time is required by this method. The method presented in [16] is based on the equivalent current injection technique. One advantage of this method is the constant Jacobian matrix that needs to be converted only once. The reactances of the distribution network feeders have not been taken into consideration by assuming that . However, distribution networks are characterized by a wide range of resistance and reactance which indicates that the

method may fail in the case of . A direct approach to obtain the distribution power flow solution was presented in [17] by same author. Two matrices, the bus injection to branch current BIBC and the branch current to bus voltage BCBV and direct matrix multiplication are used to obtain the distribution power flow solution, DLF. The solution involves direct matrix multiplication, thus large memory space is needed especially if it applied for large-scale distribution systems. Further, these matrices contain many zero elements, so the memory space is not economically utilized, and large CPU time is required. Prakash and Sydulu [18] have introduced certain modifications to previous method so that the DLF matrix is obtained directly without the formulation of either the BIBC or the BCBV matrices. However, balanced conditions are assumed, which is not the case in realistic systems. The network topology based method has also been modified to include the weakly meshed networks in [19]. In this paper a methodology for solving power flow problem for radial distribution networks is presented. Two matrices the line Primitive Impedance Matrix (PIM) and the Branch-Current Matrix (BCM) are developed to obtain the power flow solution. In this method, optimal node ordering which is necessary in the forward-backward substitution method, LU-decomposition, or Y-bus formulation and factorization are no longer needed. The proposed method has been tested on several distribution networks. This paper is organized as follows: in section II various components models are given. Section III described the solution methodology. Two case studies are explained in Section IV. Test results and discussion are given in Section V. Concluding remarks and future work are finally presented in Section VI. II.

COMPONENTS MODELING

The models of various distribution network components have a great influence on the accuracy of the results of the power flow program. This section discusses briefly the network components that were used in the presented power flow solution. A. Modeling of Unbalanced Three-Phase Line Section A four-wire distribution network is assumed in this paper as this type of systems is widely used worldwide. Fig.1 shows a three-phase line section connected between two buses and . Zaa

VA

Va Zab

Zac

VB

Zbb

VC

Vb Zbc

Zan Zbn

Zcc

Vc Zcn

VN

Znn

Fig. 1 Three-phase line section model

Vn

The parameters of the line can be found by the method developed by Carson and Lewis [6]. A 4 4 matrix that takes the effect of the self-and-mutual coupling between phases can be expressed as,

i. -connected loads Fig. 2 shows a three-phase Y-connected unbalanced load model. ILc

c Vcn b ILb

(1)

Vbn SLc

SLb n

However, it is convenient to represent (1) as a 3 3 matrix instead of the 4 4 matrix by using Kron’s method. The effect of the ground conductor is still included in the resultant matrix, that is (2)

Van

SLa

a I La

Fig.2 Unbalanced Y-connected load From this figure, the load current injections at the bus for three-phase Y-connected or single-phase connected line-toneutral can be expressed as,

Now by applying the KVL to the circuit model of Fig.1, the relationships between the bus voltages and branch currents can be simply written as, =

=

(3)

It should be noted, however, that single phase and two phase line sections are most common in distribution networks. Hence, in this research for any phase that fails to present, the corresponding row and column in (3) will have zeros entries. For example, for two phase line section with and phase, equation (3) will be reduced to, (4)

(6)

=

ii. -connected loads Referring to Fig. 3 shown below, the current injections at the bus for three-phase loads connected in or single-phase connected line-to-line can be expressed as, c Vbc

SLb

b ILbc

Also, for single phase line section, equation (3) will be reduced to,

ILab Vca

Vab S La

(5)

SLc ILca

A. Modeling of Loads Loads in distribution network can be spot loads or distributed loads or both.

a

Fig.3 Unbalanced -connected load

1. Spot Loads Equivalent current injection technique was used in this research to represent the distribution network loads. This is attributed in the first place to the nature of the loads in the distribution networks which are inherently unbalanced loads. It is assumed that all three-phase loads are connected or , and all single-phase and two-phase loads have connections between line and neutral and line-to-line, respectively. Further, a constant power model at each bus was assumed during the realization of this work.

=

(7)

2. Distributed Loads Distributed loads are assumed to be uniformly distributed along the line and lumped at point F, the midpoint of Fig. 4.

However, due to the inclusion of this factious bus, the computational speed will probably be increased. Therefore, a load transfer technique based on KVL and KCL has been used to reduce the power flow equations. In Fig. 4 shown below, represents the line impedance while represents the load. ZL/2

Bus m

AC

Bus F

ZL/2

ImF

VF

(9)

Bus n

Where, Vn

AC

Vm

From Fig. 6, the current injections due to line charging at bus can be written as,

(10)

InF

SL

III.

Fig. 4 Representation of distributed load After the load transfer is applied, the distributed loads can be represented as shown below in Fig. 5, Bus m Vm

Bus n Vn ILn

ZL

ILm SL/2

ALGORITHM DEVELOPMENT

The proposed approach is based on the development of two matrices the line Primitive Impedance Matrix (PIM) and the Branch-Current Matrix (BCM) to obtain the power flow solution. This section describes the development procedure of the proposed method. A. Equation Development Consider the radial distribution system with and -branches shown below in Fig. 7

SL/2

A Z34

Fig. 5 Load transfer of distributed loads

A

4

-buses 5

Z 45

B3

B4

3

B. Modeling of Capacitors Capacitors are assumed of Y-connected with ground conductor. The current injections are given as [20],

1

ABC

IL4

ABC

2

Z12

IL5

Z23

B1

BC

B2

SS Bus IL2 B5

Z 36

IL3 6

(8) IL6

Fig.7 6-bus distribution system C. Modeling of Shunt Capacitances Charging capacitors have a noticable influence on system’s voltage profile. In section A, a model for unbalanced threephase line section is introduced. This model can be improved by including the line shunt (charging) capacitors. A model takes into account the effect of the self and mutual capacitance is shown in Fig. 6. Bus m Va

Bus n Va

yab/2

yab/2

Vb

Vb yca/2

ybc/2

yca/2

ybc/2

Vc ycc/2

Vc ybb/2

yaa/2

yaa/2

ybb/2

Fig. 6 Shunt capacitance line model

ycc/2

The complex load at bus can be expressed in terms of the real and reactive power as, (11) Then, the corresponding current injected at the same node can be further described as, (12) The relationships between the bus current injections, branch currents, and voltage at the various system buses can be obtained by applying Kirchhoff’s current law KCL and Kirchhoff’s voltage law KVL, respectively. The branch currents can be expressed by equivalent current injections as,

3. Calculate node currents from the following equation, (13) Where is the node number, is load complex power and is the node shunt admittance 4. Calculate the branch currents vector as follows: The branch currents can generally be described in vector form as,

where

is the nodes injected currents vector,

(14) Similarly, the branches’ primitive impedance vector between buses and can be given as, (15) Now, the relationship between branch currents and bus voltages can be determined by applying the KVL, as will be given later. The step-by-step development of the BrachCurrent-Matrix BCM is extensively explained in section B.

5. Calculate the branches’ voltage drops, Where is the branch number, is current matrix of branch and is primitive impedance matrix of branch 6. Starting from the substation nodes towards the subsequent nodes, calculate the new node voltages as follows: for nodes and which are connected through branch , 7. After calculating all node voltages, calculate the maximum error between the new voltages and the voltages of the previous step.

B. Algorithm Development BCM Formulation: 1. Number the system buses from to where is the number of system buses and let 1 denotes the substation bus. 2. Label the branch currents such that the branch between buses and is . 3. Form the branch currents matrix, , as follows: i. Start from bus phase , fill the element in the branch current matrix with 1. ii. Search for bus phase C in the line section data in the receiving end buses column and determine the number of the bus that is connected to bus phase in the sending end buses column, e.g., . iii. If , fill the element in the branch current matrix with and let and go to ii, otherwise go to iv. iv. Reduce by one and check if , repeat substeps i, ii and iii, otherwise stop and go to sub-step v v. Repeat for phases and such that for phase use and ; for phase use and . Power Flow Solution: 1. Form the branch currents matrix, 2. For flat start, all nodes’ voltages are assumed to be except for the voltages of the substation nodes which are specified.

If the maximum error is less than or equal to the specified tolerance, go to step 9, otherwise continue to step 8. 8. Use new voltage values and repeat steps 3 to 7. 9. Print out the power flow results. Bus 2  A B C  A   1 0 0     B1 B   0 1 0    C 0 0 1     A  0 0 0     B2 B   0 0 0  C   0 0 0     A  0 0 0   BCM  B3 B   0 0 0    C  0 0 0      A   0 0 0    B4 B   0 0 0  C   0 0 0     A  0 0 0     B5 B   0 0 0  C   0 0 0    

IV.

Bus 3  A B C 1 0 0   0 1 0 0 0 1   1 0 0   0 1 0 0 0 1   0 0 0   0 0 0 0 0 0   0 0 0   0 0 0 0 0 0   0 0 0   0 0 0 0 0   

Bus 4  A B C 1 0 0   0 0 0 0 0 0   1 0 0   0 0 0 0 0 0   1 0 0   0 0 0 0 0 0   0 0 0   0 0 0 0 0 0   0 0 0   0 0 0 0 0 0  

Bus 5  A B C 1 0 0    0 0 0  0 0 0   1 0 0    0 0 0  0 0 0   1 0 0    0 0 0  0 0 0   1 0 0    0 0 0  0 0 0    0 0 0    0 0 0  0 0 0  

Bus 6  A B C 0 0 0   0 1 0 0 0 1   0 0 0   0 1 0 0 0 1    0 0 0   0 0 0 0 0 0   0 0 0   0 0 0 0 0 0   0 0 0   0 1 0 0 0 1  

CASE STUDIES

For any new method, it is very important to check its validity with other methods that exist in the literature. To verify the convergence characteristics and the accuracy of the proposed method, it is implemented on balanced and unbalanced distribution systems. The first test system is taken from Ref. [21] and is given below in Fig. 8 for explanation purposes.

The system nominal voltage is 12.66KV with two feeder substations, 33 bus, and 5 tie lines. The total substation loads for the base configuration are 5084.26 KW and 2547.32 KVARs. The branch data, loading, and voltage profile for the base configuration can be found in the same reference. 25 24

27

28

29

30

31

32

33

7

8

9

10

11

12

13

14

23

1 SS Bus

26

2 3

4

5

6

15

16

17

18

19 20 21 22

Fig. 8 A 33 bus distribution system The second test system was acquired from Ref.[19] and is shown below in Fig. 9. In this system, the single-phase and double-phase laterals have been lumped to form unbalanced loads for testing purposes. This system is characterized by its high ratio (app.1.8), which makes it an ill-conditioned distribution network and should be analyzed on three-phase basis rather than single-phase ones. The base values for this system are chosen to be 14.4 KV and 100 KVA. 3 3a SS Bus

5

2

5b

1a 2a 4 1b 4b

2b 1c 2c

6

4c 7

6c 7c

8 8c

Fig. 9 An 8 bus distribution system V.

TEST RESULTS

The proposed three-phase power flow solution is carried out using Matlab 7.9® environment with a tolerance of 0.0001. Two methods have been used in the tests for comparison purposes. These methods are, Method 1: The proposed method. Method 2: The method reported by Jen-Hao Teng [19]. The above two methods were compared on the basis of their performance parameters. A. Power Flow Solution of the 33-bus System The power flow solution of the 33-bus system is given in Table I. It is observed that the proposed method took three

iterations to converge for an accuracy of 0.0001. Method II took same number of iterations, however the proposed method has utilized less memory space as only dot product was performed to obtain the power flow solution. The maximum deviation is almost zero for five decimal places. B. Power Flow Solution of the 8-System The power flow results of the 8-bus system of Fig.9 are given in Table II. The method converged in three iterations. Method II took three iterations to converge too. The maximum deviations in both voltage magnitude and angle from Method 2 are 0.0001 and 0.0020, respectively. Though Method II has the same number of iterations, the memory space and, hence, the computational time taken by this method is higher than the proposed method since Method II involves the formulation of a matrix whose dimensions of to obtain the distribution power flow solution. The product of the PIM and BCM in the presented method, however, results in a matrix with dimensions of to obtain the distribution power flow solution. This means that the saving in memory space memory will be . For example, for an 11-bus system, the matrix that is involved in the power flow solution of Method 2 will have dimensions of . However, for our approach i.e. Method 1, the matrix will have dimensions of , therefore, the saving in memory space is . C. Robustness Test of the Proposed Method Divergence of the solution of the power flow problem usually occurs when certain ill-conditioned are presented in the system. As it is mentioned earlier in Section I, distribution networks with high ratio are inherently ill-conditioned power networks. Ill-conditioned often takes place when the system involves short-lines, which is the case in this paper, or even very long lines. In order to prove that the proposed method can be used for such harsh situations, a robustness test was done on the distribution network depicted in Fig. 8. Firstly, selected primitive impedances were taken from each section of the system sections at random basis. Secondly, among the selected impedances, one is divided by a factor of 10 to represent a short-line while the other is multiplied by 10 to have a very long line representation. For example, sections 2-4, and 4-5 are multiplied by 10 whereas sections 4-6 and 78 are divided by 10. Test result shows that the method converged slower than the base case by an amount of 21.5%, which is an acceptable figure in such severe conditions and the method has proven to be suitable for ill-conditioned distribution networks. VI.

CONCLUSION

An approach for solving radial distribution system power flow problem has been proposed in this paper. The method utilizes only basic circuit theory fundamentals. Unlike other traditional methods, the presented work has not utilized any Y-bus formulation or factorization. Also, the LUdecomposition has not been used in the presented study. The

approach has proven to be economically-effective as it utilizes less memory space to store the necessary data. The method is tested on balanced and unbalanced systems and the test results were found to be closed to other methods in the same field. It is suggested that the proposed method could be used for distribution network planning and expansion in which repeated power flows are required. The method could also be modified to accommodate the changes that usually happen in the existing network by modifying the PIM and the BCM matrices. Moreover, the proposed approach can easily be modified to incorporate constant current, constant impedance or exponential load models. The application of the proposed method to include large systems is in progress and results will be reported in due course.

TABLE I I LOAD FLOW RESULTS OF THE 8-BUS SYSTEM Bus Number & Phase 1-A 1-B 1-C 2-A 2-B 2-C 3-A 4-B 4-C 5-B 6-C 7-C 8-C

TABLE I LOAD FLOW RESULTS OF THE 33-BUS SYSTEM Node no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 KW-losses KVAr-losses P-supplied Q-supplied

Method 1 Node Vol.(p.u) 1.00000 0.99701 0.98288 0.97537 0.96795 0.94947 0.94594 0.93229 0.92596 0.92010 0.91923 0.91771 0.91154 0.90925 0.90782 0.90644 0.90439 0.90378 0.99649 0.99291 0.99220 0.99157 0.97930 0.97263 0.96930 0.94754 0.94497 0.93353 0.92531 0.92175 0.91759 0.91668 0.91639 210.987 143.128 3925.99 2443.13

Method 2 Node Vol.(p.u) 1.00000 0.99701 0.98288 0.97537 0.96795 0.94947 0.94594 0.93229 0.92596 0.92010 0.91923 0.91771 0.91154 0.90925 0.90782 0.90644 0.90439 0.90378 0.99649 0.99291 0.99220 0.99157 0.97930 0.97263 0.96930 0.94754 0.94497 0.93353 0.92531 0.92175 0.91759 0.91668 0.91639 210.987 143.128 3925.99 2443.13

Method 1

Method 2

(p.u.)

Angle (Deg.)

(p.u.)

Angle (Deg.)

1.0000 1.0000 1.0000 0.9839 0.9711 0.9697 0.9832 0.9652 0.9668 0.9640 0.9649 0.9683 0.9671

0.0000 -120.0 120.0 0.1830 -119.76 119.97 0.1790 -119.73 119.93 -119.74 119.92 119.96 119.96

1.0000 1.0000 1.0000 0.9839 0.9712 0.9697 0.9832 0.9652 0.9669 0.9640 0.9650 0.9683 0.9671

0.0000 -120.0 120.0 0.1830 119.76 119.97 0.1780 -119.73 119.93 -119.74 119.92 119.96 119.95

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VII. BIOGRAPHIES Salem Elsaiah (S’09) received Bsc. (Engrg) degree in Electrical and Electronic Engineering in 1999 from the University of Garyounis, Benghazi, Libya. After his graduation, he joined the department of Electrical and Electronic Engineering of the same university as an assistant lecturer. He earned Msc. Degree in Electrical Engineering specialized in the Electrical Machines and Power in 2006 from Garyounis University too. Currently, he is pursuing a Ph.D program at Michigan State University, USA. His area of interest includes distribution system design and planning, and power system reliability. (email: [email protected]) Mohammed Benidris (S’10) received Bsc. (Engrg) degree in Electrical and Electronic Engineering in 1998 from the University of Garyounis, Benghazi, Libya. After his graduation, he joined the department of Electrical and Electronic Engineering of the same university as an assistant lecturer. He earned Msc. Degree in Electrical Engineering specialized in Power System in 2005 from Garyounis University too. Currently, he is pursuing a Ph.D program at Michigan State University, USA. His area of interest includes power system reliability, and power system planning. (email: [email protected]) Joydeep Mitra (S ’94, M ’97, SM ’02) is an Associate Professor of Electrical Engineering at Michigan State University, East Lansing. Prior to this, he was Associate Professor at New Mexico State University, Las Cruces, Assistant Professor at North Dakota State University, Fargo, and Senior Consulting Engineer at LCG Consulting, Los Altos, CA. He received a Ph.D. in Electrical Engineering from Texas A&M University, College Station, and a B.Tech. (Hons.), also in Electrical Engineering, from Indian Institute of Technology, Kharagpur. His research interests include power system reliability, distributed energy resources, and power system planning. (Email: [email protected])