Investigation about Power Flow Algorithm for Distribution Network with Distributed Photovoltaic

To obtain the practical power flow algorithm for distribution network with distributed photovoltaic (PV), the change in phase and amplitude of calculated node voltage with number of iterations, and computation time of the Newton-Raphson algorithm, back/forward sweep algorithm and implicit Z bus Gaussian algorithm are obtained for 5-node circular distribution network with distributed photovoltaic. The results show that the Newton-Raphson algorithm can not converge and can not be used in the circular distribution network with PV. Both back/forward sweep algorithm and implicit Z bus Gaussian algorithm can converge and their results are consistent, but the computation time of the latter is almost 1/3 of the former. The latter is recommended for power flow calculation for annular distribution network with distributed PV.


Introduction
The renewable energy sources mainly including distributed photovoltaic are more and more widely used in the distribution network.Power flow calculation forms the basis of power system security and economic analysis.Distributed photovoltaic will introduce new problems to power flow calculation, making it difficult to apply the traditional power flow algorithm in the power system with distributed photovoltaic (PV) [1].Therefore, it needs to be further studied.
For early small-scale power systems, power flow calculation is mainly completed by professional manual calculation.Later, computers are gradually used for power flow calculation.In the early stage, the method based on nodal admittance matrix is mainly used, but the convergence of this method is poor.In order to improve the convergence, the Newton-Raphson algorithm [2][3] is proposed.In this method, the relative equations are expanded according to Taylor series and the higher order terms are discarded, which has good convergence.In order to deal with complex power system power flow calculation problems, the back/forward sweep method is put forward [4][5].The basic principle of this algorithm is to push forward from the root node to the end node to obtain the current of all branches, and in turn then push back from the root node to the end node to obtain the voltage of all nodes.After repeated iterations, a more accurate solution is obtained.To optimize power flow computation, the implicit Z bus Gaussian method is proposed [6][7], which iteratively calculates the node voltage of the system based on the superposition principle.This algorithm has advantages in dealing with the network with many fixed voltage nodes.At present, the comparison of the effect of various algorithms for calculation of power flow in distribution network with distributed photovoltaic has not been published in the literature.
To fix the problem, this work compares the error, convergence and computation time about power flow computation by the Newton-Raphson algorithm, back/forward sweep method and implicit Z bus Gaussian method for 5-node circular distribution network, and according to the results some suggestions for algorithm selection are presented.

Newton-Raphson method
The relationship between node power and voltage when the node voltage method is adopted is shown in Eq. (1).
where Q i and P i are the reactive power and active power for the i th node, respectively.j is the imaginary unit; U i means voltage at i th node; Y ik is i th row and k th column element of network admittance array; U ik means the branch voltage between the i th and k th nodes; The superscript * indicates conjugation; n means the total number of branches.For the rectangular coordinate system, voltage of the i th node can be represented by Eq. ( 2).
The node from the 1 st to M th belongs to PQ node, and its input active power and reactive power are P is and Q is , respectively.The power balance equation for general PQ node can be written as Eqs.( 3)- (4).
The node from the M+1 th to n-1 th belongs to PV node.If the input active power of the node is P is and the input voltage amplitude is U is , then each PV node satisfies Eqs. ( 7 and  are i th and k th nodes' voltage at l+1 th iteration, respectively. is the branch current between i th and k th nodes at the l th iteration, and the positive directional current is from i th to k th nodes. means impedance of branch between i th and k th nodes. The branch current of each feeder is the sum of the load current of all downstream nodes, so the branch current of the m th subnetwork can be calculated by Eq. (15).
(15) where, I Bm is the branch current; I Lm is the load current; T m is the correlation matrix between feeder branches and load currents in the m th subnetwork of the distribution network.The i th row and k th column elements of T m is defined as  0 the  branch is not associated with the  load 1 the  branch is associated with the  load (16)

Implicit Z bus Gaussian method
For distribution network with distributed PV, the power flow equation can be expressed by Eq. ( 17).I=YU (17) where, U means the node voltage vector; I means the node injection current vector; Y means the nodal admittance array of the distribution network.The system equation can also be written as Eq. ( 18). 
where I 1 and U 1 are the current and voltage vectors of the source nodes, respectively.I 2 and U 2 are the current and voltage vectors of the remaining nodes except the source nodes, respectively.If there is no constant power node in the system and I 2 is a known constant injection current, then U 2 can be directly calculated according to Eq. ( 19).     (19) When the change in U 2 during the iteration is smaller than the critical value, the solution of the equation is obtained.

Comparison of power flow algorithms
The Newton-Raphson algorithm, back/forward sweep algorithm and implicit Z bus Gaussian algorithm are applied to calculate the power flow of a 5-node circular distribution network with distributed PV shown in Fig. 1.The maximum iteration number is set to 15.The applicability of different algorithms is compared.
The changes in node voltage and phase calculated by the three algorithms with the number of iterations are shown in Figs. 2, 3 and 4, respectively.Note that in the figures θ 1 , θ 2 , θ 3 , θ 5 , U 1 , U 2 , U 3 and U 5 mean the phase and amplitude of voltage at the first, second, third, and fifth nodes, respectively.Fig. 2 shows that the phase and amplitude of the node voltage calculated by Newton-Raphson method oscillate with number of iteration and do not converge within the maximum iteration number.Therefore, the Newton-Raphson algorithm is not appropriate for the power flow computation for the circular distribution network with distributed PV.Figs. 3 and 4 show that the node voltage phase and amplitude calculated by back/forward sweep algorithm and the implicit Z bus Gaussian algorithm oscillate with number of iterations at the beginning, and tend to stable values with number of iterations, and eventually converge.Therefore, the back/forward sweep algorithm and the implicit Z bus Gaussian algorithm can be used to calculate power flow for circular distribution network with distributed PV.Obviously, back/forward sweep method converges after 12 iterations.The implicit Z bus Gaussian algorithm converges after 5 iterations, and the corresponding computation time is 12.53 ms and 4.28 ms, respectively.In conclusion, implicit Z bus Gaussian algorithm is recommended for computation of power flow in circular distribution network with distributed photovoltaic.

Conclusions
Power flow of 5-node circular distribution network with distributed PV is calculated by using the Newton-Raphson algorithm, the back/forward sweep algorithm and implicit Z bus Gaussian algorithm.
The results reveal that the Newton-Raphson method can not converge and can not be used in circular distribution network with distributed photovoltaic.Both back/forward sweep algorithm and implicit Z bus Gaussian algorithm can converge and their results are consistent, but the computation time of the latter is almost 1/3 of the former.Implicit Z bus Gaussian method is recommended for calculation of power flow for circular distribution network with distributed PV.

Figure 1 .
Figure 1.Five-node circular distribution network with distributed PV.

Figure 2 .
Change of voltage amplitude, phase with the number of iterations when the Newton-Raphson method is used.(a) Voltage amplitude (b) Phase Figure 3. Relationship between the voltage amplitude, phase and the number of iterations when the back/forward sweep algorithm is used.(a) Voltage amplitude (b) Phase Figure 4. Relationship between the voltage amplitude, phase and the number of iterations when implicit Z bus Gaussian algorithm is used.
is the Jacobian matrix.∆Wand∆Uareshown in Eqs.(12) and (13), respectively.Starting from the voltage value of the root node of distribution network at step l+1, voltage distribution is updated from the branch current at step l, and the node voltage of branch between the i th and k th nodes satisfies Eq. (14).