A method for power flow fault analysis of unbalanced distribution networks with multiple PV nodes

Based on the characteristics of the distribution network, which is radial, weakly looped and unbalanced in three phases, this paper combines the multi-port hybrid compensation technology on the basis of the previous generation back-propagation to achieve the unification of the three-phase power flow and non-ground fault calculation. The ground fault is handled separately, thus avoiding the non-convergence phenomenon caused by the zero voltage at the grounding point in the unified calculation of general power flow and fault. At the same time, in view of the situation where there are one or more PV nodes in the distribution network, a method for compensating PV nodes is proposed.
Keywords : distribution network; compensation technology; PV node

An Approach for Power Flow Analysis and Fault Analysis of Asymmetrical Distribution System with Multi-PV Node

PENG Shu-tao, LIAO Pei-jin, LI Lin, WANG Guang-ming

(Department of Electrical Engineering, Xi'an Jiaotong University,
Xi'an , 710049)

Abstract : This paper shows a uniform approach for three-phase power flow and un-grounding fault. Based on forward and backward sweep method and distribution system specialties, the approach makes use of multi-port compensation technology. In order to avoid divergence in the common uniform approach because of grounding fault, grounding fault is dealt with other method. This paper also presents a compensation method for PV node.
Key words : distribution system; compensation technology; PV node

0 Introduction
Flow fault calculation is a basic calculation of power grid. A reliable flow fault calculation method must be accurate, stable and practical. For distribution network, due to its own characteristics, such as generally operating in radial or weak ring network mode, the three-phase load and electrical parameters are unbalanced, and the branch parameters R and X are not much different. Therefore, special requirements are put forward for the flow fault calculation method of distribution network. First of all, the convergence problem is highly valued in the flow calculation of distribution network. For example, due to the large ratio of branch parameters R and X, the algorithms that were originally effective in transmission network, such as fast decoupling method, are no longer effective in distribution network. Secondly, in fault calculation, due to the existence of three-phase asymmetric components and loads, the flow fault calculation cannot adopt the single-phase calculation method, but must be three-phase flow fault calculation. In the three-phase flow fault calculation, it can be divided into phase component method and sequence component method according to the different ways of processing three phases. Due to the characteristics of distribution network itself, the phase component method is better in the flow fault calculation of unbalanced distribution system ^[1,2] .
At present, the power flow fault calculation for distribution network mainly uses the previous generation back-pushing and multi-port compensation techniques ^[3-6] . Reference [3] has pointed out the advantages of this method, but it also has certain shortcomings ^[6] . Reference [5] uses the superposition principle to calculate the fault current based on the previous generation back-pushing power flow algorithm, considering the radial characteristics of the distribution network. This method has high computational efficiency when dealing with pure radial power grids, but the article does not give a calculation method for the ring network, and the calculation formula given is for the balanced distribution network system. Reference [6] uses a systematic method to solve the problem of unified analysis of power flow and faults in the three-phase unbalanced weak ring network system of the distribution network, with good convergence and high calculation accuracy. However, this method does not mention how to deal with the PV nodes in the distribution network. At the same time, when a ground fault occurs, the voltage of the grounding node tends to zero during the iteration, so that the injected current of the node tends to infinity, affecting the convergence of this method.
In view of the above problems, based on a large number of Chinese and foreign literature, this paper provides a method to solve the PV nodes existing in the distribution network, and proposes a method to handle the fault separately according to whether it is grounded, which solves the problem of unified calculation of power flow faults. From the results of the example, the convergence, calculation speed and calculation accuracy can meet the requirements. 1 Standard fault port compensation circuit model Literature [7] points out that for any asymmetric fault, the asymmetric fault can always be separated from the power network at the fault point. Figure 1 is a schematic diagram of the fault compensation circuit separated from the power network.

The impedance of the fault compensation circuit in Figure 1 is: In this way , different faults

can be simulated by selecting the values ​​of Z _fg , Z _fa , Z _fb , and Z _{fc . The values ​​of Z fg} , Z _fa , Z _fb , and Z _fc under several typical fault forms are shown in Table 1.

Among them, ALG means single-phase grounding of phase A; BCLL means short circuit between phases BC; BCLG means grounding between phases BC; ABCLL means three-phase short circuit; ABCLG means three-phase grounding. 2 Calculation of power flow faults in weak ring networks For weak ring networks, before calculating power flow faults, each ring network must be torn into a radial shape, the nodes must be numbered, and the branches must be layered. When numbering nodes, the root node is first selected. The default root node number is 1. The numbering principle is: assuming that m nodes need to be numbered in a certain step, the new order of these m nodes is determined according to the number of nodes at the next level to which they are connected. When the number of nodes at the next level connected to some nodes is the same, the new order of these nodes is determined according to their original order. Figure 2 is used here to illustrate the method of node numbering. Assuming that node 1 has been numbered, it can be seen from the figure that nodes 2, 3, and 4 are connected to node 1. Now they need to be renumbered. It can be seen that the number of next-level nodes connected to nodes 2, 3, and 4 is 1, 2, and 3 respectively, so the numbering of nodes 2, 3, and 4 in the new order is 2, 3, and 4 (the new order is the same as the original order because Figure 2 is a network diagram that has been completed according to the numbering principle). After completing the numbering of nodes 2, 3, and 4, the numbering of all nodes in Figure 2 can be completed in the same way. After the node numbering is completed, the branches are layered according to the size of the nodes at both ends of the branches, and the branches are numbered and layered according to the principle of small node priority. Figure 2 is a completed new network diagram (only the node numbers are marked here).

2.1 Calculation of Radial Network Power Flow
First, the amplitude and phase angle of the voltage of each node in the network are initialized with the amplitude and phase angle of the root node voltage. The algorithm for solving the radial network power flow by iterative method includes four steps in each iteration. At the kth iteration:
(a) Calculation of node injection current. Starting from the root node, the injection current of each node is calculated layer by layer:

Here, Ii _is a column vector composed of the three-phase injection current of node i; Vi _is a column vector composed of the three-phase voltage of node i; Si _is a column vector composed of the three-phase injection power of node i; Yi _is the admittance sum of all grounded branches of node i.
(b) Back-pushing branch current. Back-pushing from the end of each feeder to the root node of the feeder, calculate the current of each branch, and the current calculation formula of branch l is:

Here Jl _is a column vector composed of the three-phase current of branch l. M represents all branches connected to node i, but excluding branch l.
(c) Previous generation operation. Update the node voltage layer by layer from the root node to the end of the feeder. When the voltage of node i is known, assuming that node j is connected to node i through branch l, the calculation formula for the voltage of node j is:

(d) Repeat the above (a) to (c) until the voltage correction amount of each node in two consecutive iterations is less than a certain threshold.
The above (1) to (3) provide formulas for solving the three-phase power flow of the radial network, but the role of the ring network and faults is not considered.
2.2 Compensation of ring network loops and fault loops
2.2.1 Unified calculation of ring network loops and non-grounded faults
When there are ring network loops and non-grounded fault points in the system at the same time, there are two loops in the system, one is the ring network loop in the power grid; the other is the fault loop composed of the fault compensation circuit, ground, feeder root node power supply and feeder. As shown in Figure 3.

In Figure 3, L and F represent the ring network and the fault loop; j1 _and j2 _are nodes where the ring network is torn; f1 _is a node in the system, and f2 _only appears when a non-ground fault occurs. Obviously, the ring network loop and the non-ground fault loop can be regarded as two ports, and the port injection current can be calculated according to the following formula:

Where ZLL _is the port impedance matrix of L, ZFF _is the port impedance matrix of F, and ZLF _is the mutual impedance between L and F. For three-phase calculation, they are all 3×3 matrices. The methods for solving these matrices can be found in the literature ^{[6, 7]} and will not be repeated here. _{I L} and I _F are equivalent injection currents.

2.2.2 Calculation of ground faults
The calculation of ground faults is based on the superposition principle. The ground fault is regarded as an additional injection current suddenly superimposed at the fault point under normal operating conditions. After a previous generation back calculation, the voltage and branch current of each node can be calculated. This additional current is calculated by the following formula:

Where Igf _is the additional injection current. Here, Z _gf can still be called the port impedance matrix. Its solution method is the same as the previous method for solving the port impedance matrix. Unlike other literatures, Z _{gf is not different due to different fault types. This is because the Z gf} formed in this paper has implicitly distinguished the fault types. V _gf = [V _fa V _fb V _fc ] ^T , the values ​​under several typical faults are shown in Table 1. If there are multiple faults at the same time, just expand Z _gf and V _gf accordingly, and the complexity of the program will not be increased. V _a|0| , V _b|0| , V _c|0| are the voltages at the fault point before the fault.

3 Compensation for PV nodes
PV nodes in the distribution network refer to those nodes in the distribution network that can maintain the voltage amplitude at a certain level by adjusting reactive power, such as reactive compensation nodes in the distribution network. The processing method for such nodes is: when the calculation of the weak ring network power flow fault has converged, if the voltage amplitude of the PV node obtained is not equal to the set voltage amplitude, it is necessary to inject a certain amount of equivalent injection current into the PV node, which can make the voltage amplitude of the PV node reach the preset value.
In order to obtain the equivalent injection current, the voltage unbalance of the PV node must be calculated first. In the calculation of three-phase power flow fault, the three-phase equivalent injection current must be taken into account, but for PV nodes, the predetermined voltage amplitude is often given, and the three phases are not distinguished. Therefore, only one phase is considered when calculating the voltage unbalance of the PV node here. The calculation formula is:

Where V _V is the voltage unbalance column vector of the PV node, and its dimension is equal to the number of PV nodes. V ^s is the set value of the PV node voltage amplitude, and V ^k is the voltage amplitude calculated by the power flow fault.
The second step is to find the equivalent injection current. Here, the PV node sensitivity matrix Z _V is introduced to calculate the equivalent injection current. The calculation formula is:

Where Z _V is a constant real number matrix. Its diagonal elements Zi _are the sum of the positive sequence reactances of all branches connected from node i to the root node. When there is no common branch from PV node i, j to the root node, the non-diagonal elements Zij in _{Z V} are zero; otherwise, _Zij is equal to the sum of the positive sequence reactances of the branches in the common part of the two roads formed by nodes i, j to the root node. Here _Iq is a linear approximation of the equivalent injection current. The actual injection current can be calculated using formula (9): Here δ _va , δ _vb , δ _vc are the phase angles of the three-phase voltages of the PV node. The reactive power limitation is not considered when calculating the equivalent injection current above. In general, since the adjustable reactive power of the PV node is not unlimited, it is necessary to calculate the reactive power with additional injection current, and the reactive power must not exceed the limit. The calculation formula is: If the reactive power exceeds the limit, the PV node must be converted into a PQ node, and Z _V must be re-formed and decomposed. According to the above, the calculation flow chart of the entire algorithm can be obtained, as shown in Figure 4.

4 Case Analysis
The method in this paper is used to simulate a typical system in PG&E. This system includes a 21KV root node, two main feeders, and a radial network. By opening and closing switches in different positions, the changes in the structure of the distribution network and the number of ring networks are simulated. Three types of faults are considered in the calculation: three-phase short circuit to ground (3LG), phase short circuit (LL), and single-phase grounding (SLG); four situations: in the first situation, there are no PV nodes and ring networks, in the second situation, there are 4 ring networks without PV nodes, in the third situation, there are 10 PV nodes without ring networks, and in the fourth situation, there are 10 PV nodes and 4 ring networks.

5 Conclusion
Based on the three-phase unified power flow fault calculation method of the distribution network, this paper proposes to unify the calculation of power flow and non-ground faults, and calculate the ground fault according to the superposition principle based on the shortcomings and defects of this method. This method has the same convergence as the power flow calculation of the weak ring network. On this basis, a processing method for PV nodes is given. From the perspective of the processing method, it is very similar to the processing method of unified calculation of power flow faults, does not increase the difficulty of programming, and conforms to the phenomenon of reactive power compensation in the current distribution network.