Development of Unbalanced Three-Phase Distribution Power Flow Analysis Using Sequence and Phase Components Mamdouh Abdel-Akher, Khalid Mohamed Nor, Senior Member IEEE, and Abdul-Halim Abdul-Rashid standard Newton Raphson method or its variants [1-5]. However, the three-phase Newton-Raphson method is computationally expensive for large systems due to the size of the Jacobian matrix and its fast decoupled version is sensitive for high line R/X ratios. The admittance or impedance methods [6-9] have convergence characteristics that are highly dependent on the number of the specified voltage buses in electrical network [8]. There are also unbalanced power-flow methods which consider primarily the radial structure of distribution networks. Therefore, these power-flow methods can solve only radial or weekly meshed systems such as methods given in [10-11]. The advantage of the network decomposition is that the size of the unbalanced power-flow problem is effectively reduced [12-16], and hence, large computational saving is achieved. There were two studies reported to decompose an electrical network in sequence or phase components. The development using sequence components [12-14] is limited to three-phase networks comprising of three-phase components. On the other hand, the development in phase components is limited to radial networks and cannot solve interconnected unbalanced transmission or distribution networks. Successive improvements have been presented to model the unbalanced power system elements in sequence components [12-14]. The sequence decoupled method in [14] has been used to analyze unbalanced three-phase transmission networks with accurate solution and superior performance characteristics. This method was able to handle both the transformer phase shifts, the line coupling, and voltage controlled nodes efficiently. However, the method limited to solve three-phase networks and can not be used to analyze distribution networks due to the existence of twophase and single-phase line segments which is difficult to be modeled in sequence components. In this paper, the basic sequence-decoupled power-flow method [14] is extended to solve unbalanced distribution three-phase power-flow. An unbalanced distribution network is decomposed into a three-phase network with three-phase line segments and unbalanced laterals with twophase and single-phase line segments. The unbalanced laterals that consist of single-phase and two-phase line segments are replaced by an equivalent current injection at their upstream nodes (i.e. root nodes). Hence, the main three-phase network is solved based on symmetrical components.
Abstract— This paper presents a new power-flow method for analyzing unbalanced distribution networks. In this method, an unbalanced distribution network is decomposed to: 1) main three-phase network with three-phase line segments and 2) unbalanced laterals with two-phase and single-phase line segments. The proposed method allows solving the main threephase network based on the decoupled positive-, negative, and zero-sequence networks. The unbalanced laterals are solved using the forward/backward method in phase components. The solution process involves three main steps. Firstly, in phase components, the backward step is executed to calculate an equivalent current injection for each unbalanced lateral. Secondly, the main three-phase network is solved in sequence components. The standard Newton-Raphson and fast decoupled methods are used for solving the positive-sequence network whereas the negative- and zero-sequence networks are represented by two nodal voltage equations. Finally, in phase components, the forward step is performed to update the voltages in the unbalanced laterals. The three-steps are repeated till convergence happen. Distribution network characteristics such as line coupling, transformer phase shifts, voltage regulators, PV nodes, capacitor banks, and spot or distributed loads with any type and connection are considered. Solution of unbalanced radial feeders shows that the proposed hybrid algorithm is accurate. Index Terms— distribution networks, power system modeling, three-phase power-flow and symmetrical components
U
I. INTRODUCTION
NBALANCED three-phase distribution networks are modeled in phase coordinates frame of reference. This is because the mutual inductances between different phases of an unsymmetrical transmission lines are not equal to each other. Besides the sequence networks are coupled together and cannot be broken into independent circuits and distribution systems contain multiphase unbalanced laterals. Consequently, a variety of three-phase power-flow algorithms have been developed based on phase components for solving unbalanced power systems. Some of these algorithms solve a general network structure such as Mamdouh Abdel-Akher is with the Department of Electrical Engineering, South Valley University, Aswan, Egypt. (email:
[email protected]). Khalid Mohamed Nor is with the Department of Electrical Power Engineering, Universiti Teknologi Malaysia, Johor, Malaysia. (e-mail:
[email protected]). Abdul Halim Abdul Rashid is with the Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, Malaysia (e-mail:
[email protected]).
978-1-4244-1933-3/08/$25.00 ©2008 IEEE
406
Substation
650 Voltage regulator
646
632
645
0bc
Two-phase line segment
611
684
00c
671
632 Three-phase current injection
Exact calculation of the equivalent current injections at the upstream nodes requires the solution of unbalanced laterals without any approximations. This is achieved by utilizing a simplified forward/backward method in phase components to solve the unbalanced laterals. Then, the solution of the whole network is obtained by putting the decomposed networks in a hybrid iterative scheme in both sequence and phase components. The proposed decomposed distribution power-flow combines the advantages of both the sequence decoupled power-flow and the forward/backward power-flow methods. The proposed power-flow method is validated by solving the IEEE radial feeders and the solution is compared with the results obtained from the radial distribution analysis program (RDAP) [19].
a00
Transformer abc
692
671 Switch
652
abc
675
Three-phase line segment
abc Node
Unbalanced laterals
634
abc
a0c Single-phase line segment
633
abc refers to the phases of a line segment, 0 indicate a null phase
680 The main three -phase network
Fig. 1 Decomposition of the IEEE 13 node test feeder into main threephase network and unbalanced laterals
II. POWER SYSTEM MODELS A. Line Segments and transformers: For the transformer and line segments, the models reported in [14] are used in the main three-phase network. In the unbalanced laterals solution, the phase component models are used [17].
total loads, capacitor banks, and the lines losses of that unbalanced lateral.
C. Voltage controlled nodes model One of the advantages of the proposed method is the simplicity of the PV node representation. The balanced three-phase program bus specifications are used similar to reference [14]. In some alternatives programs, it is very difficult to model the PV node such as the RDAP software.
B. Unbalanced laterals A distribution network contains many unbalanced laterals. For example, the IEEE 13 node feeder contains two unbalanced laterals plugged at bus 632 and at bus 671. A decomposed unbalanced lateral involves only two-phase levels. The first level is a two-phase circuit while the second is a single-phase circuit such as the unbalanced lateral plugged at bus 671. Consequently, an unbalanced lateral comprises of a few numbers of buses and branches. Due to this fact, unbalanced laterals represent small, simple, and independent radial distribution networks. Each unbalanced lateral has a source bus, such as bus 632 and bus 671, in which voltage should be specified to solve the unbalanced lateral having specified values for loads and capacitors. Therefore, the unbalanced laterals, that represent small or simple radial networks, can be solved properly and efficiently by a simplified forward/backward sweep algorithm.
III. DECOMPOSITION OF DISTRIBUTION NETWORKS
IV. FORWARD / BACKWARD METHOD
B. Load Model: The load can be modeled as constant power, constant current, or constant load model, or any combination of these models. The load may be connected as star or delta with any number of phases. The loads are represented by equivalent current injections. In the case of the main three-phase network, the phase current injections are transformed to their counterparts in sequence components. Distributed loads are modeled similar to reference [10].
The difficulty of the application of sequence networks for solving distribution systems can be avoided by decomposition of distribution networks into main threephase network and unbalanced laterals. Example for the decomposition of the IEEE 13 node feeder is shown in Fig. 1. The unbalanced laterals are decoupled from the main three-phase circuit and replaced by equivalent current injections. A. Main three-phase network The main three-phase network includes all power system elements having all phases (a-b-c), i.e. complete three-phase network without any single-phase and two-phase circuits. Therefore, the sequence decoupled networks, positive-, negative-, and zero-sequence networks, can be established. Hence, the sequence decoupled power-flow algorithm can be used for solving the main three-phase network. However, the solution requires calculating the injected current at the buses having unbalanced laterals such as bus 632 and 671 of the IEEE 13 bus feeder. The injected currents at an unbalanced lateral plugged at a certain bus represent the 407
The forward/backward method requires a certain numbering to the network elements. Examples of numbering schemes are branch numbering in [13]. The IEEE 13 node feeder is arranged according to the branch numbering scheme as shown in Fig. 2 and Table 1. The forward/backward sweep method [13] contains mainly three steps to solve an unbalanced distribution radial network. The three steps are: 1. Nodal current calculation: the first step is to calculate all the current injection at different buses in the feeder due to loads and capacitor banks, and any other shunt elements based on initial voltages. In the next iterations, the nodal currents are calculated using the updated voltages. 2. Backward sweep: At iteration k, starting from branches at the last layer and moving towards the branches connected to the root node, all branch currents are calculated by applying current summation. For a branch L, the branch current is calculated as follows:
Fig. 2 Forward/backward sweep solution for the IEEE 13 bus feeder k
k
k
J L _ a I j _ a Jm_a J L _ b = − I j _ b + J m_b JL_c I j _ c m∈M J m _ c (1) Where J is the current that flows on a line section, L and M refers to the set of the line sections connected to node j , I is
∑
Bus i
the current injected due to shunt elements at bus j, and k is the iteration serial. 3. Forward sweep: Nodal voltages are updated in a forward sweep starting from branches in the first layer toward those in the last layer. The updated voltage are corrected starting from the root node as follows: k
k
k
cir cu
ci r cu
ph as
e
se
Two phase circuit
Two phase level or single phase level may contain any number of nodes
si n gl e
ph a gl e sin
First bus also belongs to the main network
TABLE II ORDERING OF TABLE I ACCORDING TO ROOT NODE AND LAYER ID Branch Root Bus i Bus j Layer ID Network ID ID node ID 650 650R 1 650R* 632 2 632 633 4 Three-phase 632 671 5 0 0 main 633 634 7 network 671 692 8 671 680 10 692 675 13 632 645 3 1 Unbalanced 632 lateral 1 645 646 6 2 671 684 9 1 Unbalanced 671 684 611 11 2 lateral 2 684 652 12 2 * refers to the regulated bus
it
it
V j _ a Vi _ a J L _ a V j _ b = Vi _ b − Z branch J L _ b V j _ c Vi _ c J L _ c (2) Where Z is branch series impedance, j is the node at which the voltage is updated, i is the node at which the voltage is updated from the former layer. The forward/backward sweep method solution of the IEEE 13 bus feeder is shown in Fig. 2. The figure shows the three main steps for solving radial network. The three steps are repeated until the convergence is recognized.
single phase circuit si ng
le
sin ph
as e
cir c
ui t
gle
ph
as e
cir
cu
Distribution Network
650 650R* 632 632 632 645 633 671 671 671 684 684 692
TABLE I MODIFIED ORDERING SCHEME FOR THE HYBRID DISTRIBUTION POWER -FLOW Branch Root Bus j Layer ID Network ID ID node ID 650R 1 0 0 632 2 0 0 645 3 632 1 633 4 0 0 671 5 0 0 646 6 632 2 634 7 0 0 692 8 0 0 684 9 671 1 680 10 0 0 611 11 671 2 652 12 671 2 675 13 0 0
it
Fig. 3 General network structure of a decomposed unbalanced lateral
V. SOLUTION OF UNBALANCED LATERALS The forward/backward sweep algorithm described in the previous section is usually used for the solution of the whole radial distribution network. In this case, the feeder comprises of three-phase levels. The first level is the main three-phase network, the second is a two-phase circuit, and finally, the third level is for the single-phase circuit. 408
Therefore, the application of the forward/backward algorithm for the solution of the whole network becomes tedious due to complicated data structure required for the solution process. However, if the forward/backward sweep algorithm is applied only for solving the decomposed unbalanced laterals, the algorithm is greatly simplified due to the simplicity of the decomposed unbalanced lateral structure. Figure 3 shows a general unbalanced lateral that may be plugged at any node in the main three-phase network. The
Unbalanced laterals plugged at buses 1,2,i,...N in the main circuit Root 1 nodes
Unbalanced laterals plugged at buses 1,2,i,...N in the main circuit Root 1 nodes
N
i
2
N
i
2
Update voltages for each unbalanced lateral feeder
Calculate current injections due to unbalanced laterals
Unbalanced Lateral N
Unbalanced Lateral 3
Unbalanced Lateral 2
Step 1: Nodal current calculation Step 2: Current summation (Backward sweep)
Unbalanced Lateral 1
Unbalanced Lateral N
Unbalanced Lateral 3
Unbalanced Lateral 2
Unbalanced Lateral 1
Go to step 1 and 2
Step 3: Voltage update (Forward sweep)
Updated voltages
Injected currents
Solution of the decoupled unbalanced Laterals Forward/Backward sweep method in phase components
Transform injected currents due to unbalanced laterals to their counterparts sequence equivalents
Positive sequence network
i N 2 1 For all root nodes Combined current injection of unbalanced laterals with current injections due to three -phase loads, capacitor banks, or untransposed lines
Negative sequence network
Zero sequence network
Final specified values
Solution of the three -phase main network Sequence decoupled power flow method proposed in [14] Fig. 4 Hybrid iterative power-flow solution algorithm
unbalanced lateral mainly comprises of two levels for phases, the single-phase and two-phase levels. It also includes little number of layers, and each layer has few numbers of branches. Consequently, a simple method can be developed based on the forward/backward method to handle the simplified unbalanced lateral radial network. Usually a distribution network contains many unbalanced laterals. These unbalanced laterals are decoupled from each other, i.e. each unbalanced lateral is a standalone radial network and having its upstream root node. The application of forward/backward sweep algorithm requires the knowledge of the voltage and the angle at the root node, i.e. each unbalanced lateral should have its own slack bus. However, this condition is not satisfied because these voltages are dependent on the solution of the main threephase network. This leads to an iterative solution between the different unbalanced laterals and the main three-phase network. The first two steps in the forward/backward are performed to calculate the current injection at the root nodes for different unbalanced laterals. The third step is postponed till the main three-phase network is solved. Hence, the voltage at the root nodes become known, the forward sweep step can be performed for updating the node voltages in each unbalanced lateral. The solution in this way results in a hybrid iterative process between the main three-phase network and the unbalanced laterals in sequence and phase components respectively.
VI. HYBRID DISTRIBUTION POWER-FLOW As discussed in the previous section, the voltages at the root nodes of the unbalanced laterals depend on the solution of the main three-phase network. Therefore, the forward 409
sweep is performed after the main three-phase network solution. This results in an iterative solution between the main three-phase circuit and the unbalanced laterals solution using the forward/backward sweep method. Hence, the solution of the whole distribution network becomes hybrid solution in both sequence and phase components. A. Branch numbering scheme The branch numbering shown in Fig. 2 is for the solution of the whole distribution network using forward/backward sweep method. However, the proposed hybrid distribution power-flow solves the network partially using the forward/backward sweep method. The branch numbering is modified for the excluding the main three-phase network. The modified branch numbering scheme include, in addition to the branch and layer identification numbers, the root node identification number. This is because the unbalanced laterals can be specified easily by their root nodes, i.e. upstream nodes. As an example for branch numbering, the data structure for branches of the IEEE 13 bus feeder is arranged in Table I. In this numbering scheme, the three-phase branches are assigned zero entry for both the layer ID and the root node ID to refer to the main threephase network. The other branches that represent two-phase circuits and single-phase circuits have entries for both the layer ID and the root node ID. The structure of the decomposed networks can be obtained by ordering Table I, firstly by root node ID, and then by layer ID. Table 2 gives the decomposed network structure. The table gives the decomposed networks of the IEEE 13 node feeder that involves three-phase main network and two unbalanced laterals plugged at nodes 632 and 671.
TABLE III SOLUTION OF THE IEEE 13 NODE FEEDER
1.05 IEEE 123 Bus Voltage profile phas e A
1.04
RDAP (Exact)
Voltage p.u.
1.03 1.02 1.01 1.00 RDAP
IEEE Solus tion Proposed
0.99
Propos ed method
0.98 RG1
72
87
69
113
11
55
27
46
610
Bus ID
650 632 645 646 633 634 671 692 675 684 611 652 650
1.0000 0.9535
0.00 -2.96
0.9502 0.9243 0.9197 0.9197 0.9125 0.9180
-3.03 -3.81 -6.30 -6.30 -6.57 -6.35
0.9122 1.0000
-6.28 0.00
650 632 645 646 633 634 671 692 675 684 611 652 680
1.0000 0.9535 0.0000 0.0000 0.9502 0.9238 0.9198 0.9198 0.9126 0.9181 0.9123 0.9198
0.00 -2.96 0.00 0.00 -3.03 -3.83 -6.30 -6.31 -6.58 -6.36 0.00 -6.28 -6.30
650 632 645 646 633 634 671 692 675 684 611 652 680
0.0000 0.0000
0.00 0.00
0.0000 0.0005 -0.0001 -0.0001 -0.0001 -0.0001
0.00 0.02 0.00 0.01 0.01 0.01
-0.0001 -0.0001
0.00 0.00
1.0000 0.9972 0.9879 0.9862 0.9952 0.9759 1.0119 1.0119 1.0143
-120.00 -121.77 -121.95 -122.02 -121.82 -122.32 -122.37 -122.37 -122.56
1.0000 0.9446 0.9429 0.9410 0.9418 0.9214 0.8977 0.8977 0.8954 0.8945 0.8912
120.00 118.14 118.17 118.22 118.14 117.58 116.50 116.50 116.53 116.47 116.39
1.0000
-120.00
1.0000
120.00
Fig. 5 Voltage profile of phase A, the IEEE 123 node feeder Proposed Method: Sequence Newton Raphson
B. Solution process Figure 4 shows the proposed hybrid iterative distribution power-flow. The algorithm starts by performing the first and the second steps of the forward/backward method. Then, the current injections due to unbalanced laterals is then passed to the sequence decoupled power-flow algorithm. These currents are transformed with other current injection due to loads, capacitor banks, or current injections due to line coupling. Therefore, the final sequence current injection for buses that have unbalanced laterals is given by (3). The sequence decoupled networks are solved according the formulation in our previous paper [14]. The standard
1.0000 0.9973 0.9880 0.9863 0.9953 0.9757 1.0121 1.0121 1.0145
-120.00 -121.77 -121.95 -122.02 -121.82 -122.33 -122.37 -122.37 -122.56
1.0000 0.9447 0.9430 0.9411 0.9419 0.9211 0.8979 0.8979 0.8956 0.8946 0.8913
1.0121
-122.37
0.8979
120.00 118.14 118.17 118.22 118.14 117.57 116.50 116.50 116.53 116.47 116.39 0.00 116.50
Error=Exact-Proposed
Newton-Raphson and fast decoupled methods are used for solving the positive-sequence network whereas the negative- and zerosequence networks are represented by two nodal voltage equations. The negative- and zero-sequence networks are represented by two nodal voltage equations [14]. After solving the sequence
networks, the root voltages of the unbalanced laterals are known, and hence, the third step of the forward/backward method can be performed for updating the voltages at the rest of buses in different unbalanced laterals. The process is repeated until convergence is reached by using phase voltages mismatch criterion since the positive sequence voltage or the positive sequence power mismatch criteria will not be accurate.
0.0000 -0.0001 -0.0001 -0.0001 -0.0001 0.0002 -0.0002 -0.0002 -0.0002
0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00
0.0000 -0.0001 -0.0001 -0.0001 -0.0001 0.0003 -0.0002 -0.0002 -0.0002 -0.0001 -0.0001
0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00
-0.0002
0.00
-0.0002
0.00
The solution shows that the solution of the proposed unbalanced power flow methods based on sequence networks agree well with the RDAP solution. If the RDAP solution is referred to be the exact solution, the error shown in Table III is only in the forth figure for the voltage magnitude and the second figure for phase angle.
VII. RESULTS The IEEE 13 bus and the IEEE 123 bus radial distribution networks are used for testing the proposed unbalanced distribution power flow. The test is performed by a comparison of the numerical three-phase voltages obtained form the Radial Distribution Analysis Program (RDAP).
B. Numerical results of the 123 bus IEEE radial feeders
A. Numerical results of the IEEE 13 bus test feeder The IEEE 13 bus distribution feeder is a small distribution feeder but it provides a good test for the most common features of distribution analysis software. The IEEE 13 node feeder is modified by excluding both the regulator and the distributed load from the standard data [19]. The solution of the IEEE 13 feeder is given in Table III for the voltage magnitude and the angle for the three-phases.
The IEEE 123 bus is the most comprehensive test feeder since it contains variety of unbalanced laterals. Therefore, the IEEE 123 bus feeder presents a good test for the proposed unbalanced power flow based on sequence networks. The results of the three-phase voltages are compared with those calculated using RDAP software. The voltage profile for phase ‘a’ is given in Fig. 5. The figure
k k k I i_k n = I i_k Load _ n + I i_Distribu ted_Load_n + I i_Cap _ n + I i_Unbalanced_Lateral _ n +
Where M refers to the set of lines connected to node i 410
M
∑ ∆I i_k Line_n k =1, k∈i
n = 0 or 1 or 2
(3)
shows that both solutions have the same voltage profile for the three-phases of the IEEE 123 bus radial feeder.
VIII. CONCLUSION The paper has presented a new formulation for unbalanced distribution power-flow based on sequence components. An unbalanced distribution network is decomposed into a main three-phase circuit and unbalanced laterals. The main three-phase network is solved using the sequence decoupled power-flow algorithm. The unbalanced laterals are solved using the forward/backward sweep method. This results in a hybrid solution in both sequence and phase components. The advantage of the proposed formulation is that a complicated distribution network is decomposed to many sub-problems. One of these subproblems is basically a standard single-phase power-flow used in the sequence decoupled power-flow algorithm. The decoupling features of the proposed formulation eventually reduce the size of the distribution power-flow problem which leads to improvements in both execution time and memory requirements. REFERENCES [1] [2] [3]
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