### 2.1 Network topology and information nodes

Figure 1 illustrates the topology of a general complex PLC communication network, which is typically composed of signal transmitters, signal receivers, power lines, and network nodes. The purpose of power line channel modeling is to determine the transmission characteristic between a signal transmitter and one of multiple receivers at a given time. The network nodes are divided into internal-type nodes and termination-type nodes. The internal nodes only connect to power lines, such as *C*
_{
m
} and *C*
_{
n
} in Fig. 1, and the termination nodes connect to at least one of the loads or the signal source, such as *T*
_{1}, *T*
_{2}, *T*
_{
i
}, and *T*
_{
j
} in Fig. 1. *V*
_{
S
} and *Z*
_{
S
} are the amplitude and impedance of the signal source, respectively.

In this paper, a new concept called the “information node”, which can cover important information regarding the node voltage and current, is introduced to help describe the state of the network. Several information nodes are placed at the points as closely as possible around a network node. For example, *D*
_{
t11} and *D*
_{
t12} are the information nodes related to the signal source impedance *Z*
_{
s
} and power line *Line_*1, respectively, which connect to termination node *T*
_{1}, and *D*
_{
cn1} is the information node of power line *Line_k*, which connects to internal node *C*
_{
n
}. Thus, at least two relevant information nodes should be assigned to a network node except for the signal source, where the voltages of the signal source and at information node *D*
_{
t11} are different because of the internal impedance *Z*
_{
S
}. Therefore, an additional information node *D*
_{
s
} should be introduced between the signal source and *D*
_{
t11}, as shown in Fig. 1.

By placing all necessary information nodes into the network, the problem of determining the transmission characteristic between signal sourc*e V*
_{S} and each receiver is converted into a problem of determining the transmission characteristic between information node *D*
_{
S
} and the information node related to each receiver (e.g., *D*
_{
t22} if the receiver is located at node *T*
_{2}).

### 2.2 Network equations based on information nodes

Without loss of generality, we assume that there are *m* power lines and *n* loads in the power network in Fig. 1. According to the principle of setting the information nodes, 2 *m* and *n* information nodes are respectively linked with *m* power lines and *n* information nodes because a power line must be connected between two information nodes, whereas a load is always related to only one information node. Because the special information nodes *D*
_{
S
} and *D*
_{
t11} are related to the signal source, the total number *S* of information nodes in the network is:

Obviously, the voltage of and the current flowing out of an information node are two unknown quantities. Thus, there are 4 *m* + 2*n* + 4 unknown quantities for the entire network. In this paper, the reference direction of the node current is out of the node.

Figure 2 shows the power line *Line_k* and its two information nodes *D*
_{
cm1} and *D*
_{
cn1.}

According to transmission theory, the *V*-*I* relationships between information nodes *D*
_{
cm1} and *D*
_{
cn1} are:

$$ \left.\begin{array}{l}{U}_{cn1}={U}_{cm1}\cdot ch\left(\gamma x\right)-{I}_{cm1}\cdot {Z}_C\cdot sh\left(\gamma x\right)\\ {}-{I}_{cn1}={I}_{cm1}\cdot ch\left(\gamma x\right)-\frac{U_{cm1}}{Z_C} sh\left(\gamma x\right)\end{array}\right\} $$

(2)

where *x* is the length of the power line *Line_k*, and *γ* and *Z*
_{
C
} are the propagation constant and characteristic impedance of this power line, respectively. We can list two equations similar to Eq. (2) for each power line; therefore, 2 *m* equations can be obtained for *m* power lines.

For *Load_*1 in Fig. 1, its voltage *U*
_{
t22} and current *I*
_{
t22} must satisfy Ohm’s Law, i.e.,

$$ {U}_{t22}={Z}_{L1}\times {I}_{t22} $$

(3)

where *Z*
_{
L1} is the load impedance of *Load_*1. Thus, *n* equations can be obtained for *n* loads.

Suppose that there are *k* information nodes that are related to a network node *C*
_{
n
}, as shown in Fig. 3.

According to the basic circuit theory, we can obtain *k* independent equations for this network node as:

$$ \left.\begin{array}{l}{U}_{cn1}={U}_{cn2}=\cdot \cdot \cdot ={U}_{cn k}\\ {}{I}_{cn1}+{I}_{cn2}+\cdot \cdot \cdot +{I}_{cn k}=0\end{array}\right\} $$

(4)

For a network with *m* power lines, *n* loads, and one signal source, there must be 2 *m* + *n* + 1 information nodes, and each node is related to one and only one of the network nodes (*Ds* is excluded). Thus, 2 *m* + *n* + 1 independent equations can be listed for all network nodes.

As shown in Fig. 4, for information nodes *D*
_{
t11} and *D*
_{
S
}, which are related to signal source *V*
_{
S
}, the following two equations are deduced:

$$ \left.\begin{array}{l}{U}_S={U}_{t11}-{I}_{t11}\times {Z}_S\\ {}{I}_S=-{I}_{t11}\end{array}\right\} $$

(5)

where *Z*
_{
S
} is the internal impedance of signal source V_{S}.

Therefore, for a power network of *m* power lines and *n* loads, 2 *m* + *n* + 2 *m* + *n* + 1 + 2 = 4 *m* + 2*n* + 3 independent equations can be obtained based on the above derivations. These equations can be expressed in matrix form as follows:

$$ \left[\begin{array}{cccc}\hfill {a}_{(1)(1)}\hfill & \hfill {a}_{(1)(2)}\hfill & \hfill \cdots \hfill & \hfill {a}_{(1)\left(4 m+2 n+4\right)}\hfill \\ {}\hfill {a}_{(2)(1)}\hfill & \hfill {a}_{(2)(2)}\hfill & \hfill \cdots \hfill & \hfill {a}_{(2)\left(4 m+2 n+4\right)}\hfill \\ {}\hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \hfill & \hfill \vdots \hfill \\ {}\hfill {a}_{\left(4 m+2 n+3\right)(1)}\hfill & \hfill {a}_{\left(4 m+2 n+3\right)(2)}\hfill & \hfill \hfill & \hfill {a}_{\left(4 m+2 n+3\right)\left(4 m+2 n+4\right)}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {U}_S\hfill \\ {}\hfill {U}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {U}_{2 m+ n+1}\hfill \\ {}\hfill {I}_S\hfill \\ {}\hfill {I}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {I}_{2 m+ n+1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill 0\hfill \end{array}\right] $$

(6)

Supposing the coefficient matrix of Eq. (6) is *A*, we can know that all elements in *A* are only related to the network topology and component parameters. Because the 4 *m* + 2*n* + 3 equations are independent, the rank of coefficient matrix *A* is

As mentioned, there are 4 *m* + 2*n* + 4 unknown variables for the entire network, whereas there are 4 *m* + 2*n* + 3 independent equations in the homogeneous linear equations. Therefore, the ratio of any two unknown variables can be calculated easily by solving Eq. (6). Thus, the voltage transmission characteristics between the signal transmitter and the receiver are:

$$ {H}_i(f)=\frac{U_{Zi}}{U_S} $$

(8)

where *U*
_{
Zi
} is the voltage of the information node related to *Load*_*i*.

By solving Eq. (6), the theoretical expression for the voltage transmission characteristic between the signal transmitter and the receiver can be easily calculated using a conventional software such as MATLAB. Then, the key factors that affect the channel transmission characteristic and the influence law can be conveniently analyzed. This method is applicable to a network with arbitrary topology.

There are few ring topologies and no mesh topology in smart grids till now. However, in IEEE1547, part 4.1.4 “Distributed resources on distribution secondary grid and spot networks” proposes ring topologies and mesh topologies in distribution networks in the future with distributed generation accessed. In this situation, the proposed method will be able to solve the problems that there is no effective channel modeling method for the application of PLC in a network with ring and mesh topologies.