1. Network Connectivity, Load Flow and State Estimation
Network connectivity analysis is carried out using an augmented incidence matrix and augmented bus
admittance matrix operations. The standard bus incidence matrix and bus admittance matrix based
Jacobi analysis is used for the load flow calculations and State Estimation.
Anyhow, if it is planned to develop the total network from the beginning, then it is most appropriate
to use the bus impedance matrix and the step-by-step development process of the bus impedance
matrix. It shall be noted that, unless the network building is carried out from the beginning it is
impossible to develop the bus impedance matrix from the bus admittance matrix through matrix
inversion.
1.1
Network Connectivity, Load Flow and Power System State Estimation Design.
This is the core of the ADMS which needs detailed design analysis. The advanced network analysis
techniques are used in this process to develop a connectivity model, load flow model and state
estimation model.
1.1.1 Network Connectivity Module
The network connectivity are identified into tree cut-sets and extracted from the network in the usual
connectivity analysis process. In this analysis a different approach is used as the minimum cut set
calculation is lengthy as well as must be carried out every time when a switch status is changed.
Therefore, traditional calculation is resource intensive. In this Network connectivity analysis, a
modified version of the network incidence matrix is used. This is an innovation in this development.
In this methodology, the Modified Incidence Matrix, which represents the connectivity of network
branches to nodes as shown below is developed.
𝐴𝑛×𝑚
𝑎11
=[ ⋮
𝑎𝑛1
⋯ 𝑎1𝑚
⋱
⋮ ]
⋯ 𝑎𝑛𝑚
1,
0,
element 𝑖 is connected to node 𝑗
element 𝑖 is not connected to node 𝑗
(1)
Where
𝑎𝑖𝑗 = {
𝑛 =number of elements
𝑚 = number of nodes
This differs from the standard incidence matrix as in the standard network incidence matrix, the
orientation of each branch is represented by the sign of the connectivity index. In this matrix if the
network element is connected to the branch, matrix holds a value of 1 in the corresponding position
and otherwise zero.
The modified bus impedance matrix for the connectivity analysis is developed in the form of a identity
matrix of the size of the elements as shown below.
𝑦 = 𝐼𝑚×𝑚
(2)
Now the modified network bus admittance matrix, 𝑌𝑏𝑢𝑠 is developed as,
𝑌𝑏𝑢𝑠(𝑛×𝑛) = 𝐴𝑛×𝑚 ∙ 𝑦𝑚×𝑚 ∙ 𝐴𝑇𝑚×𝑛
(3)
In this matrix, the off-diagonal element which are either 1 or 0 and when 1 represent connectivity of
the node to the other node. If the two nodes are not connected, then the corresponding off-diagonal
element is 0. The diagonal elements which hold a positive non-zero number represent the number of
elements connected to the particular node.
Now, the modified power injection matrix is developed such as to hold the power feeding nodes with
number 1 and all the other nodes with 0.
𝑝1
𝑃=[ ⋮ ]
𝑝𝑚 𝑛×1
(4)
Where
𝑝𝑖 = {
1,
0,
if branch 𝑖 is energised
if branch 𝑖 is not energised
Now the energized element matrix can be developed as
[𝐵]𝑚×1 = [𝑌𝑏𝑢𝑠 ]𝑛×𝑛 [𝑃]𝑛×1
(5)
Matrix 𝐵 is modified by a recursive process to make all the positive integers in the matrix 1 and all the
zeros to remain as it is. There are no negative integers as both 𝑃 and 𝑌are positive. If the recursive
equation is developed at the nth recursion, then,
Say
[𝐵]𝑛×1
𝑏1
⋮
= 𝑏𝑖
⋮
[𝑏𝑛 ]𝑛×1
(6)
Then
1,
(𝑝𝑖 )𝑛+1 = {
0,
if (𝑏𝑖 )𝑛 > 0
if (𝑏𝑖 )𝑛 < 0
(7)
Equation (2) and equation (3) are recursively used until matrix P stabilizes. Once the matrix stabilizes
it gives the elements which are energized.
For example, the network which is shown in Figure 2 below will result in an incidence matrix as shown
in the Table 1 below. When this incidence matrix is recursively iterated as described above, the
resulting column vector indicating the energized elements is as shown in table 3. The recursive
operation is carried out using the MATLAB code as shown in table 2. The elements energized as shown
in table 3 when graphically represented looks as shown in Figure 3.
Figure 2 Single Line Diagram of the example netwrok for connectivity analysis-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
1
1
2
1
1
3
0
0
4
5
1
1
1
1
6
7
8
1
1
1
1
1
1
1
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
Table 1 – Modified Incidence Matrix for the example network used for connectivity analysis
The calculation is as shown in the table below done in MATLAB.
while 1
y=eye(size(A,2));
Ybus=A*y*A';
B=Ybus*P;
for i=1:size(A,1)
if B(i)>0
B(i)=1;
else
B(i)=0;
end
end
sum(P)
if sum(P)==sum(B)
break;
end
Table 2 – Program Code of the Network Analysis
Branch
ID
1
Energised
or not
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
1
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
Table 3 – Branch Connectivity Vector
Figure 3 – Results of the connectivity Analysis shown in the single line diagram
This particular algorithm developed by the design team is coded into the ADMS through an above
discussed outsourced coding contractor with proper testing and verification cycle.
1.1.2 Load-flow calculation[2], [3].
The data measurements are obtained from the R-LBS and ARs installed in the network. All the primary
substation busbar voltages are also assumed to be available. All the loads of the transformers 𝐿𝑡𝑟𝑓
are calculated by the equation shown below using the customer mix and the individual customer
category load curves organized in to load curve matrix 𝐿𝐶 and customer participation matrix 𝐶𝑃 for
the transformers.
Load curve of the family of transformers is calculated as,
[𝐿𝑡𝑟𝑓 ]𝑛_𝑡𝑟𝑓×24 = [𝐶𝑃]𝑛_𝑡𝑟𝑓×𝑐𝑢𝑠_𝑐𝑎𝑡 [𝐿𝐶]𝑐𝑢𝑠_𝑐𝑎𝑡×24
(8)
Where,
𝐿𝑡𝑟𝑓 = Transformer Load matrix where each row represents the load curve of each
transformer;
[𝐶𝑃]𝑛_𝑡𝑟𝑓×𝑐𝑢𝑠_𝑐𝑎𝑡 = Customer participation matrix where each row represent the percentage of
load of each customer category in each transformer.
[𝐿𝐶]𝑐𝑢𝑠_𝑐𝑎𝑡×24 = Load Curve matrix where each row represents the load curve of each customer
category
𝑛𝑡𝑟𝑓 = Number of transformers
𝑐𝑢𝑠𝑐𝑎𝑡 = Number of customer categories (Usually 9) such as General Purpose, Domestic
0-30, Domestic 60-90, Domestic 90-120, Domestic 120-180, Domestic>180,
Industrial, Hotel, Government, Religious, etc
The standard load flow is carried out for the radial network using the dc load flow equations coupled
with the incident matrix where in this instance the directionality is incorporated as shown below.
𝐴𝑛×𝑚
𝑎11
=[ ⋮
𝑎𝑛1
⋯ 𝑎1𝑚
⋱
⋮ ]
⋯ 𝑎𝑛𝑚
(9)
Where
1,
𝑎𝑖𝑗 = { 0,
−1,
element 𝑖 direction is towards node 𝑗
element 𝑖 is not connected to node 𝑗
element 𝑖 direction is away from node 𝑗
𝑛 =number of elements
𝑚 = number of nodes
Standard load flow equations are developed for this purpose and incorporated to the ADMS as a
separate module.
These calculations are indicated here for completion.
Here the straightforward process of calculating load flow using standard Jacobi method is described.
In the distribution load flow problem, all the busbars are load busbars and therefore the issue of the
inclusion of generator busbars and voltage control busbars will not occur. Here, the calculation starts
from assuming a set of voltages from the busbars at which the complex power injections are known.
Iteratives the next set of voltages are calculated from the present set of voltages until the set of bus
voltages converge.
The ith row of the matrix equation 𝐼 = 𝑌𝑏𝑢𝑠 𝑉 can be written as
𝑛
𝐼𝑖 = ∑ 𝑌𝑖𝑗 𝑉𝑗
𝑗=1
Now from the definition of complex power injection at ith row 𝑆𝑖 = 𝑉𝑖 𝐼𝑖∗ we can write,
𝐼𝑖 =
𝑆𝑖∗
𝑉𝑖∗
Gives
𝑛
𝑆𝑖∗
= ∑ 𝑌𝑖𝑗 𝑉𝑗
𝑉𝑖∗
𝑗=1
Now this can be expanded in extracting the diagonal items
𝑛
𝑆𝑖∗
= 𝑌𝑖𝑖 𝑉𝑖 + ∑ 𝑌𝑖𝑗 𝑉𝑗
𝑉𝑖∗
𝑗=1
𝑗≠𝑖
Which can be rearranged as
𝑛
1 𝑆𝑖∗
𝑉𝑖 =
− ∑ 𝑌𝑖𝑗 𝑉𝑗
𝑌𝑖𝑖 𝑉𝑖∗
{
𝑗=1
𝑗≠𝑖
(10)
}
Equation (10) can be used to form an iterative solution so that the new voltages can be calculated
from the know voltages as,
𝑛
𝑉𝑖𝑛𝑒𝑤
1 𝑆𝑖∗
𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠
=
− ∑ 𝑌𝑖𝑗 𝑉𝑗
𝑌𝑖𝑖 𝑉𝑖∗
{
𝑗=1
𝑗≠𝑖
(11)
}
Therefore, the process of calculating the bus voltages is as below.
𝑆∗
1. Calculate the currents of each node by using 𝐼𝑖 = 𝑉𝑖∗ for all buses.
2. Now calculate ∑𝑛𝑗=1 𝑌𝑖𝑗 𝑉𝑗
𝑖
𝑗≠𝑖
3. Calculate the new bus voltage by using, 𝑉𝑖𝑛𝑒𝑤 =
1
𝑆𝑖∗
{
𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠
∗
𝑌𝑖𝑖 𝑉
𝑖
𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠
− ∑𝑛𝑗=1 𝑌𝑖𝑗 𝑉𝑗
}
𝑗≠𝑖
4. Compare the voltage values calculated from the iteration with the previous values to check
whether the solution is converged.
1.1.3 Estimation of states[1], [4], [5]
The states of this system can be considered as the bus voltages. Unlike in HV networks, the bus
voltages are not measured by voltage transformers. In distribution network, the buses are not well
formed busbar connection points in grid substations. Busses are the network nodes identifying the
equipment connection points and the network branching points and hence, the measurement of
voltages in all these network nodes is costly and not practical. There are two equipment installed for
other purposes which got voltage measurements. The Remote Operation Load Break Switches (R-LBS)
which are installed for load breaking and network isolation purposes and Auto-Reclosures (AR) which
are installed for the network switching purposes have voltage measurement incorporated. Further
the boundary metering points equipped with CT-PT units have the voltage reading of the node.
Usually, the number of buses where the voltages are measured is much lesser in number than the
total number of nodes.
In power system state estimation equation is given as.
𝑥 𝑒𝑠𝑡 = [[𝐻 𝑇 𝑅 −1 𝐻]−1 𝐻𝑇 𝑅−1 ]𝑧 𝑚𝑒𝑎𝑠
(10)
Where
𝑥 = Power system states
𝑧
𝑚𝑒𝑎𝑠
= Measurements,
𝐻 = Mapping function of measurements to each state
𝑅 = Covariance matrix of measurement error
The mapping function of 𝑁𝑆 states to 𝑖 𝑡ℎ measurement, if expressed as,
𝑓𝑖 (𝑥1 , 𝑥2 , ⋯ , 𝑥𝑁𝑆 ) = 𝑓𝑖 (𝒙) = ℎ𝑖1 𝑥1 + ℎ𝑖2 𝑥2 + ⋯ + ℎ𝑖𝑁𝑠 𝑥𝑁𝑠
Then we can say the 𝑁𝑆 states can be mapped to 𝑁𝑚 measurements as,
𝑓1 (𝑥)
𝑓 (𝑥)
𝒇(𝒙) = [ 1
⋮ ]
𝑓𝑁𝑚 (𝑥)
= [𝐻]𝑁𝑚 ×𝑁𝑠 [𝑥]𝑁𝑠 ×1
(11)
𝑁𝑚 ×1
Where
[𝐻] = 𝑁𝑚 × 𝑁𝑆 matrix
𝑁𝑚 = Number of measurements
𝑁𝑠 = Number of states
The measurements are arranged as a column vector as,
𝒛𝑚𝑒𝑎𝑠
𝑧1𝑚𝑒𝑎𝑠
𝑧 𝑚𝑒𝑎𝑠
= 2
⋮
𝑚𝑒𝑎𝑠
[𝑧𝑁𝑚 ]
(12)
𝑁𝑚 ×1
And, the covariance matrix can be written as,
𝜎12
𝜎22
𝑅=
(13)
⋱
[
𝜎𝑁2𝑚 ]
It shall be noted that if the number of measuremets is more than the numebr of states then the
maximum likelyhood function as defined above can be used. In the case if the number of states and
number of measurements are the same then a reduced version of the maximum likelyhood function
as shown below can be used.
𝑥 𝑒𝑠𝑡 = [𝐻]−1 𝑧 𝑚𝑒𝑎𝑠
(14)
In the case of actual number of measurements is lesser than the states, which is the most likey case
for distribution system estimates, the estimate equation become undterministic. In this case a set of
pseudo set of measurements are created to make the number of measurements equal to the number
of states.
The measurements of the LECO network are the voltages and the pseudo measurements are the
current injections at nodes. Therefore the load flow equation which related current and voltage can
be rewritten for the estimation purposes as,
𝐼 = 𝑌𝑏𝑢𝑠 𝑉
Which gives
𝑉 = [𝑌𝑏𝑢𝑠 ]−1 𝐼
(15)
If the voltage vector is split for measured and unmeasured voltages, then
𝑉
𝑃
[ 𝑚] = [
𝑅
𝑉
𝑄 𝐼
][ ]
𝑆 𝐼𝑝
(16)
This can be re-arranged using the matrix partition to derive a function between the measurement
voltages and pseudo currents and the unmeasured voltages and unassumed currents as;
𝑄𝑆 −1
𝑉
[ ] = [ −1
𝐼
𝑆
−1
𝑃 − 𝑄𝑆 −1 𝑅
]
−𝑆 −1 𝑅
𝑉𝑚
[𝐼 ]
𝑝
(17)
This is similar to equation 𝑥 𝑒𝑠𝑡 = [𝐻]−1 𝑧 𝑚𝑒𝑎𝑠 where
𝑄𝑆 −1 𝑃 − 𝑄𝑆 −1 𝑅
(18)
]
𝑆 −1
−𝑆 −1 𝑅
This equation can be used to do the state estimation. The covariances for the assumption can be used
as the covariance of the data set used for the transformer load estimation.
[𝐻] = [
An alternative solution to this exists as, the direct determination of the bus impedance matrix through
the sequential development of the network. This is used as the modality of network modification in
order to eliminate the burden of determining the 𝑆 −1 every time the network is modified. This is also
used for the state estimation in the networks to create an overdetermined situation where the power
of all the buses are known and the voltages of several buses are known. In such situation the bus
impedance matrix needs to be built using the step by step method and the state to measurement
mapping function as in equation (11) can be used as the impedance bus matrix.
𝑉 = 𝑍𝐼
Which can be expanded as
𝑛
𝑉𝑖 = 𝑍𝑖1 𝐼1 + 𝑍𝑖2 𝐼2 + ⋯ + 𝑍𝑖𝑛 𝐼𝑛 = 𝑍𝑖𝑖 𝐼𝑖 + ∑ 𝑍𝑖𝑗 𝐼𝑗
𝑗=1
𝑗≠𝑖
And by the definition of complex power,
𝑆𝑗 = 𝑉𝑗 𝐼𝑗∗
Gives
𝑆𝑗∗
𝐼𝑗 = ∗
𝑉𝑗
Substituting in the above equation we get
𝑛
𝑆𝑗∗
𝑆𝑖∗
𝑉𝑖 [1 − 𝑍𝑖𝑖
]
=
∑
𝑍
𝑖𝑗 ∗
|𝑉𝑖 |2
𝑉𝑗
𝑗=1
𝑗≠𝑖
If we assume that for the |𝑉𝑖 | we use the steady state load flow value, then
1
𝑉𝑖 =
[1 − 𝑍𝑖𝑖
𝑛
𝑆𝑗∗
∑ 𝑍𝑖𝑗 ∗
𝑆𝑖∗
𝑉𝑗
𝑗=1
]
|𝑉𝑖 |2 𝑗≠𝑖
This function is used as the mapping function of the state estimation process.
The covariance matrix is also required to be developed, if the number of measurements are greater
than the states in any of the above circumstances ( 𝑁𝑚 > 𝑁𝑠 ).
From equation (8) we can write the load value of the ith transformer at nth hour as
𝑐𝑢𝑠_𝑐𝑎𝑡
𝐿𝑡𝑟𝑓,𝑖,𝑛 = ∑ 𝐶𝑃𝑖,𝑚 𝐿𝐶𝑚,𝑛
𝑚=1
Now if the standard deviation of customer category percentages of the 𝑚𝑡ℎ customer category is
𝜎𝐶𝑃,𝑚 and the standard deviation of the Load Curve of 𝑚𝑡ℎ customer category for all hours is 𝜎𝐿𝐶,𝑚
Then the standard deviation of the Load of the 𝑖 𝑡ℎ transformer at 𝑛𝑡ℎ hour is,
𝑐𝑢𝑠_𝑐𝑎𝑡
𝜎𝐿𝑡𝑟𝑓 ,𝑖 = √ ∑
𝑚=1
2
2
𝜎𝐿𝐶,𝑚
𝜎𝐶𝑃,𝑚
2
2
𝜎𝐿𝐶,𝑚
+ 𝜎𝐶𝑃,𝑚
(19)
We can assume that this is the standard deviation of pseudo currents used for the state estimation.
The standard deviation of the measured voltages can be extracted from the measurement equipment
error data. Therefore, the covariance matrix can be calculated from equation 13 as,
𝝈2𝑉𝑚
𝑹𝑉𝑚
[𝑹 ] = [
𝐼𝑝
𝑛
𝝈2𝐿𝑡𝑟𝑓,𝑖
]
(20)
Therefore, in an over determined situation the state estimation can be done by using equation (10),
(18) and (20).
These are the mathematical algorithms used for the connectivity analysis, load flow and the state
estimation of the network. These algorithms are coded as Operational Procedures on the SQL
database, which can be invoked by the APIs coded in the API platform or otherwise the procedures on
the software.
1.1.4 Standard Theory of the development of Bus Impedance Matrix in step-by-step method.
The development of the bus impedance matrix starts from connecting an impedance (say 𝑍𝑎 ) to the
reference bus, creating bus no ①. As the reference bus is considered as the ‘zero’ bus the matrix will
look as,
[𝑉1 ] = [𝑍𝑎 ][𝐼1 ]
Now we can connect impedance 𝑍𝑏 to bus ① by creating bus ②. This will introduce a column and
row to the matrix as,
𝑉
𝑍
[ 1] = [ 𝑎
𝑉2
0
0 𝐼1
][ ]
𝑍𝑏 𝐼2
This matrix is considered as the elementary bus impedance matrix.
There are four cases of the modification of the above elementary 𝑍𝑏𝑢𝑠 matrix.
Case 1 – Adding 𝒁𝒄 connected between a new bus P and the reference bus
Case 1-Addition of a bus connected to the reference bus
The bus impedance matrix will be inserted with a new row and a column as shown below
𝑉𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙
𝑍𝑜𝑟𝑔
=
[
⋯
𝑉𝑝
]
⋯
[0
⋯
..
⋯
0
⋮ 0
⋮ ..
⋮ 0
∙ ⋯
⋮ 𝑍𝑐 ]
Case 2 – Adding 𝒁𝒄 connected between existing bus K and new bus P.
Case 2 - Addition of a bus connected to an existing bus
A new row and a column is appended to the matrix with the kth row vector and kth column vector of
the original impedance matrix. The diagonal element is inserted with the value of 𝑍𝑘𝑘 + 𝑍𝑐 .
Case 3 – Adding an impedance 𝒁𝒄 between the reference bus an existing bus K.
Case 3 – Addition of a branch between existing bus and reference bus
Insert bus P as case 2 and short circuit bus P and bus 0 by making 𝑉𝑃 = 0.
This can now be rewritten by matrix partitioning as;
And can be expanded to apply Kron’s Reduction as,
[𝑉1−𝑁 ]𝑁×1 = [𝑍𝑜𝑟𝑔 ]
[𝐼 ]
𝑁×𝑁 1−𝑁 𝑁×1
+ [𝑍𝑐𝑜𝑙−𝑘 ]𝑁×1 ∙ 𝐼𝑃
0 = [𝑍𝑟𝑜𝑤−𝑘 ]1×𝑁 [𝐼1−𝑁 ]𝑁×1 + (𝑍𝑘𝑘 + 𝑍𝑐 )𝐼𝑃
Form the second equation
𝐼𝑃 = −
1
[𝑍
][𝐼 ]
𝑍𝑘𝑘 + 𝑍𝑐 𝑟𝑜𝑤−𝑘 1−𝑁
And substitution of this result in the first equation gives
[𝑉1−𝑁 ] = [𝑍𝑜𝑟𝑔 ][𝐼1−𝑁 ] −
[𝑉1−𝑁 ] = {[𝑍𝑜𝑟𝑔 ] −
1
[𝑍
][𝑍
][𝐼 ]
𝑍𝑘𝑘 + 𝑍𝑐 𝑐𝑜𝑙−𝑘 𝑟𝑜𝑤−𝑘 1−𝑁
1
[𝑍
][𝑍
]} [𝐼1−𝑁 ]
𝑍𝑘𝑘 + 𝑍𝑐 𝑐𝑜𝑙−𝑘 𝑟𝑜𝑤−𝑘
Which gives
[𝑍𝑛𝑒𝑤 ] = [𝑍𝑜𝑟𝑔 ] −
1
[𝑍
][𝑍
]
𝑍𝑘𝑘 + 𝑍𝑐 𝑐𝑜𝑙−𝑘 𝑟𝑜𝑤−𝑘
This can be expanded to result
𝑍11
[𝑍𝑛𝑒𝑤 ] = [ ⋮
𝑍𝑁1
⋯
⋱
⋯
𝑍1(𝑁+1)
𝑍1𝑁
1
⋮ ]−
[ ⋮ ] [𝑍(𝑁+1)1
𝑍
+
𝑍
𝑘𝑘
𝑐
𝑍𝑁𝑁
𝑍𝑁(𝑁+1)
The general element of row h and column i can be written as,
𝑍𝑛𝑒𝑤(ℎ,𝑖) = 𝑍ℎ,𝑖 −
𝑍ℎ(𝑁+1) 𝑍(𝑁+1)𝑖
𝑍𝑘𝑘 + 𝑍𝑐
⋯ 𝑍(𝑁+1)𝑁 ]
Case 4 – Adding an impedance 𝒁𝒄 between an existing bus K and existing bus J
Case 4 – Addition of a branch between two existing buses
First a Row and Column is inserted to the bus impedance matrix as shown below.
Where
𝑍𝛽 = 𝑍𝑗𝑗 + 𝑍𝑘𝑘 − 2𝑍𝑗𝑘 + 𝑍𝑐
Now the newly added row and column is eliminated by the above-described Kron’s reduction as
𝑍𝑛𝑒𝑤(ℎ,𝑖) = 𝑍ℎ,𝑖 −
𝑍ℎ(𝑁+1) 𝑍(𝑁+1)𝑖
𝑍𝑗𝑗 + 𝑍𝑗𝑘 − 2𝑍𝑗𝑘 + 𝑍𝑐
Sequential application of these four cases any network can be built.