Distribution systems are diﬀerent from transmission systems in a number of respects, such as the branch ratio, magnitudes of and and most importantly the typically radial structure. Due to these diﬀerences, a number of power ﬂow solution methods have been developed to account for the speciﬁc nature of distribution systems and most widely used are the backward/forward sweep methods [15, 16]. Matpower includes an additional three AC power ﬂow methods that are speciﬁc to radial networks.
When solving radial distribution networks it is practical to number branches with numbers that are equal to the receiving bus numbers. An example is given in Figure 41, where branches are drawn black. Furthermore, the oriented branch ordering [17] oﬀers a possibility for fast and eﬃcient backward/forward sweeps. All branches are always oriented from the sending bus to the receiving bus and the only requirement is that the sending bus number should be smaller than the receiving bus number. This means that for branch –. The indices of the sending nodes of branches are stored in vector such that .
As usual, the supply bus (slack bus) is given index 1, meaning that branch indices should go from 2 to which is the number of buses in the network. Introducing a ﬁctitious branch with index 1 and zero impedance, given with dashed black line in Figure 41, the number of branches becomes equal to the number of buses .
For the example of Figure 41 vector is the following

and it oﬀers an easy way to follow the path between any bus and bus 0. If we consider bus 4, the path to the slack bus consists of following branches: branch 4, since considered bus is their receiving bus; branch 3, since ; branch 2, since ; and branch 1, since .
The representation of branch , connecting buses and , is given in Figure 42, where it is modeled with its serial impedance . At both ends there are load demands and , and shunt admittances and comprised of admittances due capacitance of all lines and shunt elements connected to buses and

The voltage calculation procedure with the Current Summation Method is performed in 5 steps as follows [15, 16].
 (4.16) 
 (4.17) 
 (4.18) 
 (4.19) 
the procedure is ﬁnished. Otherwise go to step 2.
The voltage calculation procedure with the Power Summation Method is performed in 5 steps as follows [17].
 (4.20) 
 (4.23) 
For each node, besides the known admittance , we deﬁne the admittance as the driving point admittance of the part of the network fed by node , including shunt admittance . We also deﬁne an equivalent current generator for the part of the network fed by node . The current of this generator consists of all load currents fed by node . The process of calculation of bus voltages with the admittance summation method consists of the following 5 steps [18].
 (4.24) 
 (4.29) 
The methods explained in the previous three subsections are applicable to radial networks without loops and PV buses. These methods can be used to solve the power ﬂow problem in weakly meshed networks if a compensation procedure based on Thevenin equivalent impedance matrix is added [15, 16]. In [17] a voltage correction procedure is added to the process.
The list of branches is expanded by a set of ﬁctitious links, corresponding to the PV nodes. Each of these links starts at the slack bus and ends at a corresponding PV bus, thus forming a loop in the network. A ﬁctitious link going to bus is represented by a voltage generator with a voltage magnitude equal to the speciﬁed voltage magnitude at bus . Its phase angle is equal to the calculated phase angle at bus .
A loop impedance matrix is formed for the loops made by the ﬁctitious links and it has the following properties
As an illustration, in Figure 41 there a two PV generators at buses 5 and 7. The ﬁctitious links and loops orientation are drawn in red. The Thevenin matrix for this case is

First column elements are equal to PV bus voltages when the current injection at bus 5 is 1 p.u. and . Bus voltages can be calculated with the current summation method in a single iteration. By repeating the procedure for bus 7 one can calculate the elements of second column.
By breaking all links the network becomes radial [16] and the three backward/forward sweep methods are applicable. Since all links are ﬁctitious, only the injected reactive power at their receiving bus is determined [17] by the following equation
 (4.30) 
which is practically an increment in reactive power injection of the corresponding PV generator for the current iteration.
The incremental changes of the imaginary part of PV generator current can be obtained by solving the matrix equation
 (4.31) 
where
 (4.32) 
In order to ensure phase diﬀerence between voltage and current at PV generators in [19] it was suggested to calculate the real part of PV generator current as
 (4.33) 
In such a way the PV generator will inject purely reactive power, as it is supposed to do. Its active power is added before as a negative load.
Before proceeding with the next iteration, the bus voltage corrections are calculated. In order to do that, the radial network is solved by applying incremental current changes at the PV buses as excitations and setting . After the backward/forward sweep is performed with the current summation method, the voltage corrections at all buses are known. They are added to the latest voltages in order to obtain the new bus voltages, which are used in the next iteration [17].