The state space model of a generic dc grid with n nodes, m branches and where p VSCs are connected, will be derived in this section. In this thesis, cables or overhead transmission lines are both modelled as Π-sections. As an example, Figure 4.5 shows a dc cable rep- resented as a Π-section connected between the dc nodes j and k. The nodes are electrical points where branches and shunt elements (such as the equivalent capacitors Cbj and Cbk
connected to the nodes j and k, respectively) are connected. A node is characterized by its voltage. For example, in the figure, ej and ekare the voltages of the nodes j and k, respec-
tively. The series elements which interconnect the nodes are called branches. For instance, in the figure it is shown the branch bi (which is composed by a series connection of the re- sistance Rbiand the inductance Lbi) which interconnects the nodes j and k. The branches
are characterized by the current which flows from one node to the other corresponding node. In the figure, the currents ij and ikrepresent current injections from external sources,
such as converters.
Figure 4.5: A cable connected between the nodes j and k modelled as a π section
the following equation Cbj dej dt =−ibi+ ij (4.35a) Cbk dek dt = ibi+ ik (4.35b) Lbidibi dt =−Rbiibi+ ej− ek (4.35c) which can be generalized as
Cde dt =−T Ti b+ Qi (4.36a) Ldib dt =−Rib+ Te (4.36b)
where e is the vector of voltages of the n dc node defined as
∆e = [e1 e2 ... en]T (4.37)
ibis vector of currents through the m branches, defined as
∆ib = [ib1, ib2 ... ibm] (4.38)
iis the vector of currents injected by the p VSCs connected to the dc grid, defined as
∆i = [i1, i2 ... ip]. (4.39)
Cis the capacitance matrix and the following rules define its jk-th element cjk =
0 if j 6= k
Ceqj if j = k (4.40)
In this case, the indices j and k are related to the dc node number, and their maximum values are n. Ceqj is the equivalent capacitor connected to the dc node j. The equivalent
capacitor is given by
Ceqj = Cbj1+ Cbj2+ ... + Cbjh+ Cjvsc (4.41)
where the subscript h represents the number of cables connected to the dc node j and Cjvsc
is the VSC capacitor connected to the node j, if any is connected to it. Then, the matrix C is a diagonal matrix whose size is n × n. L is the inductance matrix and its jk-th element is defined as follows
ljk =
0 if j 6= k
Lbj if j = k (4.42)
In this case, the indices j and k are related to the branch number, and their maximum values are m. Lbj is the equivalent inductor of the branch bj, which comes from the Π-model of
the cable. Then, the matrix L is a diagonal matrix whose size is m × m. Finally, the jk-th element of the resistance matrix R is defined as:
rjk =
0 if j 6= k
In this case, the indices j and k are related to the branch number, and their maximum values are m. Rbj is the equivalent resistor of the branch bj, which comes from the Π-model of
the cable. Then, the matrix R is a diagonal matrix whose size is m × m.
Tis the so-called incidence matrix [53], which gives information about the interconnection of the different branches and nodes. The information needed to build matrix T is “from” which node “to” which node the branch current flows. Then, the jk-th element of the matrix T is defined as follow:
tjk =
+1 if k corresponds to the “from” bus of the branch j −1 if k corresponds to the “to” bus of the branch j
0 if k does not correspond to any of the buses to where the branch j is connected
(4.44)
In this case, the index j is related to the branch number and its maximum value is m. The index k is related to the dc node number, and its maximum value is n. Then, the size of the matrix T is m × n.
Qis called the current injection matrix in this thesis, since it gives information on to which nodes the VSCs inject current. The following defines the jk-th element of Q
qjk =
0 if the VSC k is not connected to the node j
1 if the VSC k is connected to the node j (4.45) where qjk is the jk-element of the matrix Q. In this case, the index j is related to the
dc node number and its maximum value is n, while the index k is related to the VSC “identification” number and its maximum value is p. Then, the matrix Q is a diagonal matrix whose size is n × p.
As a result, the state space model of the dc grid is given by dxg
dt = Agxg + Bgig (4.46a)
e = Cgxg (4.46b)
which is a linear system. Equation (4.46), in terms of small signals, is d∆xg dt = Ag∆xg + Bg∆i (4.47a) ∆e = Cg∆xg (4.47b) where ∆xg = ∆e ∆ib , Ag = 0n×n −C−1TT L−1T L−1R , Bg = C−1Q 0m×p , Cg = [In×n 0n×m]