CHAPTER 3

Power System Network Matrices—2

3.1 Introduction

In the previous chapter, we have discussed the methods for formulating Z_Bus either directly or through the inversion of Y_Bus. These topological methods, though effective for small-sized networks, have the following disadvantages when applied to large networks such as a real power system:

• For a large-sized power system, the dimension of Y_Bus is very high. Inversion of such large size matrices may have computational difficulties.
• Network changes such as outage of lines, addition of lines and disconnection of sources are frequent. For these changes, the topology or the structure of the network is not the same and we shall have to formulate of network matrices from the beginning again. This procedure is unnecessary and is rather wasteful.
• The computational effort in constructing Z_Bus by adding one element at a time is much greater than that required to construct Y_Bus^. However, Z_Bus furnishes more information regarding the network, as is evident from the non-zero elements (zero sparsely) present in the Z_Bus matrix. Further, if the network contains open circuits such as those in the zero sequence network of the system, then some zero elements may be present. Otherwise Z_Bus is never sparse.

Bus admittance matrix has been used in power flow studies (discussed in Chapter-4) because of its sparsity advantages, whereas the bus voltages are explicitly expressed in terms of bus currents using the bus impedance matrix in short-circuit studies (Chapter-6).

This chapter presents an algorithm for building up the bus impedance matrix in a step-by-step procedure (adding one element at a time). In this building up algorithm, modifications required in Z_Bus can be easily incorporated in accordance with the network changes.

3.2 Partial Network

A partial power system network is a linear passive inter-connected network with all passive elements being thoroughly connected to form a complete network. However, some or all the buses are available outside as shown in Figure 3.1.

 

 

Fig 3.1 n-Bus Partial Power System Network

 

The performance equations in the bus-frame of reference can be written as:

and represent n-number of system equations for an n-bus power system.

The matrix Z_Bus is a symmetric square matrix of dimension (n 3 n) and is given below:

 

 

The performance equations are:

 

 

Procedure for finding the elements of Z_Bus:

To find the diagonal elements Z_ii, we inject I_i = 1 p.u current at the i^th bus and open all remaining buses such that all I_k^'s are zeros for k = 1, 2,…, n but k ≠ i (See Figure 3.2). Since Ii = 1 p.u current, measurement of the i^th bus voltage with respect to the reference bus directly gives the value of Zii as:

 

 

Fig 3.2 Determination of Diagonal Elements in Z_Bus

 

 

To find the off-diagonal elements Z_ik, inject 1 p. u current at the k^th bus and keep the remaining buses open such that all I_i^'s are zeros for i = 1, 2,…, n but i ≠ k. Since I_k = 1 p. u the measurement of the i^th bus voltage (I = 1, 2, …, n but i ≠ k) shall give directly the value of Z_ik as:

 

 

Fig 3.3 Determination of Off-Diagonal Elements in Z_Bus

 

 

The following sections present the algorithm for building up ZBus by adding only one element at a time, in a step-by-step process. This procedure can be programmable and has the advantage of not requiring a complete rebuilding of the ZBu_s for any modification of the network.

3.3 Case Studies in Z_Bus ALGORITHM

For an existing n-bus partial connected power system network, the Z_Bus would have been previously formulated. Let this be designated as Z_Bus-_Old and is given by Equation (3.1), which is repeated below for convenience.

 

 

Whenever a new element is added or removed, the matrix modifies Z_Bus-Old to Z_Bus-New. We shall now discuss modifications to the Z_Bus algorithm in the following cases. First, we consider the modifications in Z_Bus when an element is added to the existing network.

Case-1: Whenever a branch (or twig) element is added, either a new bus (or node) is created. This increases the dimension of Z_Bus-New to (n + 1 3 × n + 1), i.e, by one row and one column, over Z_Bus-Old. The elements of the new row and column need to be determined.

Case-2: Whenever a link element is added, the dimension of Z_Bus remains the same. However, some of the elements of Z_Bus-Old matrix related to the link elements need to be modified to obtain Z_Bus-New.

Each of the above two cases can be further subdivided into four types:

Type-1: The added element is a branch element between a new bus and the reference bus.

Type-2: The added element is a branch element between an existing old bus (other than the reference bus) and a new bus.

Type 3: The added element is a link element between the reference bus and an existing old bus.

Type-4: The added element is a link element between two existing old buses other than the reference bus.

 

 

Fig 3.4 Various Cases in Z_Bus Algorithm

 

Further, the added element may or may not have mutual coupling with the already existing elements in the partial network. Figure 3.4 depicts all the above cases in building up the algorithm for Z_Bus.

3.4 Algorithm for Formation of Bus Impedance Matrix — No Mutual Coupling between the Elements

This section presents the four type-modifications in the partial network and describes the consequent modifications in the Z_Bus. A simple case of 2-bus partial network is chosen for better understanding of the algorithm. However, the modifications are later generalized for large-sized n-bus power system.

3.4.1 Type-1 Modification

The performance equation for a 2-bus partial network Figure (3.5) can be written as:

 

 

 

Fig 3.5 Type-1 Modification (No Mutual Coupling)

 

A branch element with self-impedance Z_s is added between the new bus (3) and the reference bus (0). Since the added element is a branch, the size of Z_Bus increases from 2 × 2 to 3 × 3 and Equation (3.5) is modified to:

 

 

To find the diagonal element Z₃₃, induce I₃ = 1 p. u current into the third bus, and open buses 1 and 2 (I₁ and I₂ = 0).

 

 

To find the off-diagonal elements, we first induce I₁ = 1 p.u (I₃ and I₂ = 0) and later I₂ = 1 p. u (I₃ and I₁ = 0) and measure voltage V₃. Since current in the buses cannot affect the voltage V₃, we can write the off-diagonal elements as:

Now, the modified Z_Bus is:

 

 

Type-1 modification for the existing n-bus partial network is given below.

For the added branch with impedance Z_s between the new bus (k) and the reference bus (0), as:

 

 

3.4.2 Type-2 Modification

The branch element with self-impedance Z_s is added from the new bus 3 to the old bus 2.

With reference to Figure 3.6, the bus voltages are:

 

 

 

Fig 3.6 Type-2 Modifications

 

 

Substituting (3.11) in (3.12) we obtain:

 

 

Rearranging the above equation,

 

 

Equations (3.10), (3.11) and (3.14) give the new matrix as:

 

 

where Z₁₃ = Z₁₂, Z₃₁ = Z₂₁, Z₂₃ = Z₂₂, Z₃₂ = Z₂₂, Z₃₃ = Z₂₂ + Z_s

 

 

In the generalized form the above Equation (3.16) can be rewritten for the added branch with impedance Z_s between the new bus k and the old bus j as:

 

3.4.3 Type-3 Modification

Let a link element with self-impedance Z_S be added between bus 2 and reference bus 0 as shown in Figure 3.7. This can be treated as an extension of Type-2 modification by introducing a new bus 3. However, bus 3 is eliminated later by setting V₃ = 0 (node elimination technique). The modified performance equation can be written as:

 

 

where Z₁₃ = Z₁₂, Z₃₁ = Z₂₁, Z₂₃ = Z₂₂, Z₃₂ = Z₂₂, Z₃₃ = Z₂₂ + Z_S

Node elimination technique

Set V₃ = 0 by closing the switch in Figure 3.7. By setting V₃ = 0 in equation in Equation (3.21),

 

 

 

Fig 3.7 Type-3 Modification

 

Substituting the value of I₃ in (3.10)

 

 

By rearranging the terms in the above equation,

 

 

and the values of I₃ in Equation (3.11), we get

 

 

By rearranging the terms in the above equation,

 

 

Equations (3.23) and (3.24) can be written in the matrix form, as:

 

 

Let Equation (3.25) be written in a more generalized form. Assume that an element with self-impedance Z_s is added between the old bus k and the reference bus in an n-bus partial power system network.

 

 

Any general term in the Equation (3.26) can be written as:

 

3.4.4 Type-4 Modification

A link element with self-impedance Z_s connects old bus 2 to old bus 3 as shown in Figure 3.8. Referring to the figure, we can write performance equations as:

 

 

 

Fig 3.8 Type-4 Modification

 

The voltages of the buses 2 and 3 can be written as:

 

 

or

Z₃₁ I₁ + Z₃₂ (I₂ + I₄) + Z₃₃ (I₃ – I₄) = Z₂₁ I₁ + Z₂₂ (I₂ + I₄) + Z₂₃(I₃ – I₄) + Z_S I₄

Rearranging, the above equation

0 = (Z₂₁ – Z₃₁₎ I₁ + (Z₂₂ – Z₃₂) I₂ + (Z₂₃ – Z₃₃) I₃ + (Z₂₂ + Z₃₃ – Z₂₃ – Z₃₂ + Z_S) I₄

or,

 

 

where Z_x = (Z₂₂ + Z₃₃ – Z₂₃ – Z₃₂ + Z_S)

I₄ in the matrix form can be written as

 

 

Separating I₄ terms in the matrix Equation (3.27) can be written as:

 

 

Substituting the value of I₄ from Equation (3.29) in Equation (3.30),

 

 

Extracting Z_Bus-New from the above equation, it can be written as:

 

 

Let Equation (3.31) be written in a more generalized form. Assume an element with self-impedance Z_s be added between the old buses i and k in an n-bus partial power system network.

Let a link element with self-impedance Z_s be added between two existing old buses, say k and l

 

 

where, Z_X = Z_S + Z_ii + Z_kk – Z_ik – Z_ki.

Any general p^th row—q^th column element in the modified matrix is:

 

3.4.5 MATLAB Program for Z_bus Formation

Students may use MATLAB function Z_bus to verify the results of numerical problems. While executing the program the input data is entered under the matrix variable “nw_data”.

It is required to take the reference bus as 0.

 

Example 3.1

Obtain the bus impedance matrix for the power system network shown in Figure 3.9(a) when the line L₅ is not connected between buses 1 and 3. Take reactance of each line as j0.2 p. u.

Solution:

Consider the graph for the network in Figure 3.9(b), that is, without L₅ line.

Step 1: Add branch 1–2. This is a Type-1 modification.

Step 2: Add branch 2–3. This is a Type-2 modification.

 

 

Step 3: Add link between buses 2 and 3. This is a Type-4 modification. With a fictitious bus l the Z_Bus is:

 

 

 

Fig 3.9(a) Power System Network for Example 3.1

 

 

Fig 3.9(b) Graph of the Network shown in Figure 3.9(a)

 

where,

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus l is:

 

 

Step 4: Add link L₃. This is a Type-3 modification. With a fictitious bus l the Z_Bus is:

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus l is:

 

 

Note: As the reference bus should be 0, observe buses are renumbered in the MATLAB program.

Example 3.2

If the line L₅ in Figure 3.9(a) is added between the buses 1 and 3, modify the Z_Bus.

Solution:

The line L₅ is another link added across buses 1 and 3. This is Type-3 modification. With a fictitious bus l the Z_Bus is:

 

 

 

The modified Z_Bus obtained by removing the fictitious bus l is:

 

 

Note: As the reference bus should be 0, observe buses are renumbered in the MATLAB program.

Note: The student is advised to refer to Section 3.6 to understand numerical examples that involve modifications in Z_Bus, where certain elements are removed.

Example 3.3

If the line L₂ is removed between the buses 2 and 3 in Figure 3.9(a), modify the Z_Bus.

Solution:

If the line L₂ in the figure is removed the modified Z_Bus can be obtained by adding a link in parallel with the element such that the impedance of the link added equals to the negative of the impedance of the element to be removed.

The actual Z_Bus of the network is:

 

 

Now, add a link across 2–3 with impedance j0.2 p. u. Type-4 modification with a fictitious bus l, the Z_Bus is:

 

 

The modified Z_Bus by removing the fictitious bus l is:

 

 

Example 3.4

The positive sequence impedance of a power system network is shown in Figure 3.10(a). All reactance are in per unit system. Obtain the Z_Bus matrix by building an algorithm.

Solution:

The graph for the positive sequence impedance network is shown in Figure 3.10(b).

Step 1: Add branch 0–1, this Type-1 modification gives:

 

 

Step 2: Add branch 1–2, this Type-2 modification gives:

 

 

Step 3: Add branch 2–3, this Type-2 modification gives:

 

 

Step 4: Add branch 3–4, this Type-2 modifications gives:

 

 

Step 5: Add link 0–4, this Type-3 modification gives:

 

 

Fig 3.10(a) Power System Network for Example 3.4

 

 

Fig 3.10(b) Graph for Figure 3.10(a)

 

Now, the modified Z_Bus obtained by removing the fictitious bus is:

 

 

Example 3.5

The zero sequence impedance of a power system network is shown in Figure 3.11(a). All reactance are in per unit. Obtain the Z_Bus matrix by building an algorithm.

Solution:

 

 

Fig 3.11(a) Zero Sequence Network for Example 3.5(a)

 

The graph for the zero sequence impedance network is shown in Figure 3.11(b)

Step 1: Add branch 0–1, this Type-1 modification gives:

Step 2: Add branch 1–2, this Type-1 modification gives:

 

 

Step 3: Add branch 2–3, this Type-2 modification gives:

 

 

Fig 3.11(b) Graph of the Network

 

Step 4: Add branch 3–4, this Type-2 modification gives:

 

 

Step 5: Add link 1–4, this Type-3 modification gives:

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus is:

 

 

Example 3.6

Obtain the bus impedance matrix of the power system network shown in Figure 3.12(a). Take reactance of each line as j0.3 p.u.

 

 

Fig 3.12(a) Power System Network for Example 3.6

 

Fig 3.12(b) Graph for the Power System Network

Solution:

The graph for the network is given in Figure 3.12(b).

Step 1: Add branch L₁ between buses 0 and 1, this Type-1 modification gives:

Step 2: Add branch L₂ between buses 1 and 2, this Type-2 modification gives:

 

 

Step 3: Add link L₃ between buses 1 and 2, this Type-4 modification gives:

 

 

where,

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus l is:

 

 

Step 4: Add link L₄ between buses 0 and 2, a Type-3 modification gives:

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus is:

 

 

Example 3.7

If the line L₁ between the buses 0 and 1 is removed in Figure 3.12(a), modify the Z_Bus.

Solution:

If the line L₁ in the figure is removed, the modified Z_Bus can be obtained by adding a link in parallel with the element such that impedance of the link added equals to the negative of the impedance of the element to be removed.

Actual Z_Bus of the network

Add the link –j0.3 p. u between buses 0 and 1, a Type-3 modification gives:

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus is:

 

 

 

Fig 3.12(c) Adding of a Link Element to Remove Branch

 

Example 3.8

If the line L₂ between the buses 1 and 2 is removed in Figure 3.12(a), modify the Z_Bus.

Solution:

If the line L₂ in the figure is removed the modified Z_Bus can be obtained by adding a link in parallel with the element such that the impedance of the link added equals to the negative of the impedance of the element to be removed.

Actual Z_Bus of the network

Add link –j0.3 p. u between 1 and 2, a Type-3 modification gives:

 

 

Now, the modified Z_Bus by removing the fictitious bus is:

 

 

Example 3.9

If the line L₃ between the buses 1and 2 is removed in Figure 3.12(a), modify the Z_Bus.

Solution:

If the line L₃ in the figure is removed, the modified Z_Bus can be obtained by adding a link in parallel with the element such that the impedance of the link added equals to the negative of the impedance of the element to be removed, as there is no mutual coupling between the elements. This is an example of removal of Type-4 modified element.

The actual Z-Bus of the network,

By adding link –j0.3 p. u between buses 1 and 2, we get

 

 

Now, the modified Z_Bus obtained by removing the fictitious bus is:

 

 

Example 3.10

If the line L₄ between the buses 0 and 2 is removed in Figure 3.12(a), modify the Z_Bus.

Solution:

If the line L₄ in the figure is removed, the modified Z_Bus can be obtained by adding a link in parallel with the element such that the impedance of the link added equals to the negative of the impedance of the element to be removed.

The actual Z-Bus of the network,

By adding link –j0.3 p. u between buses 0 and 2, we get

 

 

Now, the modified Z_Bus by removing the fictitious bus is:

 

 

3.5 Algorithm for the Formation of Z_Bus— CONSIDERATION OF MUTUALLY COUPLED ELEMENTS

In this section for the network changes we modify the bus impedance matrix by taking into account the mutual impedance.

3.5.1 Type-1 and Type-2 Modifications

In these cases, a new branch p-q is added as shown in Figure 3.13 to a partial (already assembled) network.

 

 

Fig 3.13 Addition of a Branch to a Partial Network

 

The performance equation for the modified network with the new branch p-q is

 

 

The network consists of bilateral passive elements, therefore Z_qi = Z_iq (q = 1, 2,…,n). Now we shall determine the elements of Z_Bus.

Determination of off-diagonal elements (Z_iq):

The elements Z_iq can be determined by injecting a current I_i = 1 p. u. at each i^th bus except at the q^th bus and calculating the voltage at the q^th bus with respect to the reference node r as shown in Figure 3.13.

Since all other bus currents are zero except at i^th bus, we can write:

 

 

Assume that the element p-q is mutually coupled with a group of elements indicated by x-y. The currents in the elements of the network can be written in terms of primitive admittances and the voltages across the elements. Thus,

 

 

where,

i_pq is the current through the added element.

v_pq is the voltage across the added element.

i_xy is the current through the already existing element in the partial network.

v_xy is the voltage across of the already existing element xy.

y_pq,pq is the self-admittance of the added element.

y_pq,xy is the vector of mutual admittance between p-q and all other elements x-y with which it has mutual coupling.

From Figure 3.13,

 

 

Further, from Equation (3.36)

 

 

The current in the added branch, i_pq, is zero because the current source is connected between the bus i and the reference. However, the voltage v_pq, across p-q, is not zero due to mutual coupling.

From Equation (3.38),

 

 

from the above,

 

 

Substituting Equation (3.39) in Equation (3.37),

 

 

By substituting Equation (3.35) in the above equation the off-diagonal elements are given by:

 

 

Determination of the diagonal element (Zqq):

To calculate Z_qq, we inject I_q = 1 p. u. current and set all other bus currents as zeros (I_i = 0 p. u. for i = 1, 2,…, n).

It can be noted from Figure 3.13,

 

 

Now measure E_q with respect to reference. By using Equation (3.34) the bus voltages are:

 

 

Substituting for ipq in Equation (3.38), we obtain

 

 

Substituting for vpq in Equation (3.37), we obtain

 

 

By substituting the Equation (3.41) in the above equation the diagonal element is given by:

 

3.5.2 Type-3 and Type-4 Modifications

Let the added element p-q be a link. A voltage source e_l is connected in series with the added element as shown in Figure 3.14. This creates a new fictitious node l which will be eliminated later. The value of e_l is such that the current through the fictitiously added branch element is zero i.e ipq = ipl = 0

 

 

Fig 3.14 Addition of a Link to a Partial Network

The performance equation for the partial network with the added element p-l and the series voltage source e_l is:

 

 

Determination of off-diagonal elements:

In the above equation E₁, E₂,…, E_n are measured voltages with respect to the reference bus. Where as E_l is the voltage of the l^th bus with respect to bus q. Therefore,

 

 

The elements Z_il, i fi l can be determined by injecting 1 p. u. current at the i^th bus and setting all other bus currents as zeros and then calculating e_l with respect to the q^th bus.

For I_i = 1 p. u., and for other bus currents zero in Equation (3.43), the bus voltages directly give impedance values as:

 

 

The voltage of the series source is:

 

 

From Equation (3.38) the ipl is:

 

 

Since ipq = ipl = 0

the value of vpl from the above equation is:

 

 

Taking the above equation can be written as:

 

 

Substituting Equation (3.47) in Equation (3.46),

 

 

Substituting Equation (3.45) in the above equation,

 

 

Determination of the diagonal element:

The value of Zll can be determined by injecting 1 p. u. current at the l^th bus and then determining the voltage of bus l with respect to bus q.

For I_l = 1 p. u., and for other bus currents zero in Equation (3.43), the bus voltages directly give impedance values as:

 

 

where, E_k (k = 1, 2,…, n) are the voltages with respect to the reference bus. The value of Zll can be found out directly by computing e_l. The current in the element p-l is

 

 

The current in terms of primitive admittances and voltages across the element are:

 

 

Since y_pl,xy = y_pq,xy and y_pl,pl = y_{pq,p q}, it follows that,

 

 

Substituting Equation (3.50) in Equation (3.46), we get

 

 

Substituting Equation (3.49) into the above equation,

 

 

Here, the summary of equations for formation of the bus impedance matrix is given as per the type modification defined earlier in Section 3.4.

3.5.3 Summary of Formulas

In this section, we summarize all types of modifications in Z_Bus.

Case-1: Mutually coupled elements are absent

For the case when mutually coupled elements are absent, the following relations are valid:

 

 

Substitution of above relations in Equations (3.40) and (3.42) yields:

Type-2 Modification

 

 

and

 

 

Type-1 Modification

If p is the reference bus, then E_P = 0.

From Equation (3.35),

 

 

Therefore,

 

 

Type-4 Modification

From Equations (3.48) and (3.51),

 

 

and

 

 

Type-3 Modification

If p is the reference bus then,

 

 

and

 

 

Elimination of l^th row and l^th column modifies the bus impedance matrix element as:

 

 

Case-2: Mutually coupled elements are present

The summary of formulas for this case is as given in the following table:

Type Modification	Addition of an element to the network	Modified ZBus elements
Type-1	A branch is added from a new bus q to the reference bus having mutual coupling with link between bus x and bus y
Type-2	A branch is added from a new bus q to an old bus p having mutual coupling with link from bus x to bus y.
Type-3	A link is added from an old bus q to a reference bus having mutual coupling with link from bus x to bus y.	By eliminating fictitious bus l,
Type-4	A link is added from an old bus q to another old bus p having mutual coupling with link from bus x to bus y.	By eliminating fictitious bus l

3.6 Modifications In Z_Bus FOR CHANGES IN THE NETWORK

When the elements are added, the procedure presented in Section 3.5 can be used. In a power system, certain elements may have to be removed in situations such as tripping of generators, transformers and transmission lines. This section presents modifications in Z_Bus for the case when elements are removed or when the impedance value of an element is changed.

Case-1: Uncoupled Case

a) Procedure for removal of an element

If the element that has to be removed is not coupled to any other element, then the procedure for modifying Z_Bus is rather a simple one. The modified Z_Bus can be obtained by adding a link element in parallel with the element, such that its impedance value is negative and equal to the impedance value of the latter. This type of network changes is applicable to Type-3 and Type-4 modifications.

b) Impedance value of an uncoupled element is changed

If the impedance value of an element is changed, then the modified Z_Bus can be obtained by adding a link element in parallel to the element. The effective impedance of both the parallel elements should be equal to the desired impedance value of the element.

Case-2: Coupled Case

Modifying Z_Bus for the case when a mutually coupled element is to be removed or its impedance value has to be changed may be illustrated with the following derivation. Since the element is coupled with some other elements in the network, it is required to consider changes in the bus currents and voltages throughout the network.

The performance equations for the partial network are given by:

The i^th bus voltage from the above equation is:

 

 

Upon removal of one element or due to change in the impedance value of a coupled element, the matrices of the network, equation (3.52) modifies to:

The i^th bus voltage now changes to:

 

 

It is to be noted that, when a mutually coupled element is removed from the network, only those Z_Bus matrix elements corresponding to the element removed get affected, while the rest of the matrix elements remain the same. However, all bus voltages and currents change in the network, and in order to get the desired bus voltages in accordance to the said network changes, we provide suitable increments to the bus currents retaining the same Z_Bus.

Following the above discussion,

 

 

From the above, the i^th bus voltage is:

 

 

where, ∆I_k change in the k ^th bus current.

Now, let a current source of 1 p. u. be connected at the j^th bus and the other buses be kept open-circuited.

i.e., I_j = 1 p. u.

and

 

 

Suppose an element p-q which is to be removed (or its impedance value is to be changed) is mutually coupled to an element say x-y. Changes in the element currents are given by:

 

 

where, ∆I_k = ∆I_pq     for k = p

 

 

and ∆I_k = 0 (for k = 1, 2,…, n; k ≠ j)

Substituting Equations (3.55) and (3.56) in Equation (3.54), we obtain the i^th bus voltage as:

 

 

Let the group of subscripts p-q and x-y be denoted as a and b, as shown:

 

 

By substituting the above-defined subscripts into Equation (3.57), its condensed form is:

 

 

In view of the primitive admittance matrix, the performance equation (3.58) can be written as:

 

 

Y'_before is primitive admittance matrix before removing the element

Y'_after is primitive admittance matrix after removing the element

The subscripts of (Y'_before — Y'_after) are cd.

 

 

Fig 3.15 Example for Formation of Y'_before and Y'_after

 

Formation of primitive impedance matrices before and after removal of elements

Consider the power system graph shown in Figure (3.15). Self and mutual reactances are marked in the figure.

The primitive impedance matrix before elimination of element between buses 1–3 is:

 

 

The primitive impedance matrix before elimination of element between buses 1–3 is:

 

 

The voltage vector in the Equation (3.59) is:

 

 

Using Equation (3.58),

 

 

Substituting Equations (3.60) and (3.61) in Equation (3.59), we get

 

 

Rearranging the terms in the above equation, we obtain

 

 

On simplification of the above,

 

 

where,

Substituting Equation (3.62) in Equation (3.58), we get

 

 

From the above equation the elements of the impedance matrix can be obtained as:

 

 

The following numerical problems explain all the modifications in Z_Bus. They consider the mutual effect and removal. Answers to the problems can be verified by executing the MATLAB program.

 

Example 3.11

A 3-bus power system network is shown in Figure 3.16(a). Obtain the bus impedance matrix by using building algorithm. Take bus (1) as reference. The self and mutual reactances are indicated in the figure.

Solution:

The graph of the power system is shown in Figure 3.16(b). Consider 1–2, 2–3 as branches and 1–3 as a link.

Step 1: Add branch 1–2, a Type-1 modification gives

 

 

 

Fig 3.16(a) Power System for Example 3.11

 

 

Fig 3.16(b) Graph of the Power System for Example 3.11

 

Step 2: Add branch 2–3, a Type-2 modification gives

 

 

Step 3: Add link 1–3, which is mutually coupled with link 2–3. This is a Type-3 modification.

 

 

The modified Z-bus obtained by removing fictitious bus is:

 

 

Example 3.12

For the power system in Example 3.11,

(a) Obtain the bus incidence matrix [A], (b) Primitive impedance matrix[z], (c) Primitive admittance matrix [y] = [z^–1], (d) Y_Busby singular transformation method, (e) Obtain Z_Bus by taking the inverse of Z_Bus.

Compare the z_Bus with the result of Example 3.11.

Solution:

With reference to the graph in Figure 3.16(b),

1. Bus impedance matrix

    

   

    

2. Primitive impedance matrix

    

   

    

3. Primitive admittance matrix

    

   

    

4.  

   

    

5.  

   

    

Z_Bus obtained in both the examples is same.

Example 3.13

Obtain the bus impedance matrix for the power system network shown in Figure 3.17(a) by the step-by-step method. The self and mutual reactances indicated in the figure are per unit values.

Solution:

The graph for the power system is shown in Figure 3.17(b). Consider the elements 1—3, 1—2 and 2—4 as branches and others as links.

Step 1: Add branch 1—2, this Type-1 modification gives:

 

 

Step 2: Add branch 1–3, which is mutually coupled with 1–2. This is also a Type-1 modification.

 

 

 

Fig 3.17(a) Power System for Example 3.13

 

 

Fig 3.17(b) Graph of the Power System of Example 3.13

 

 

Step 3: Add link 2–3, this is Type-4 modification.

 

 

Now, the modified Z-Bus by eliminating the fictitious bus,

 

 

Step 4: Add branch 2–4, this is Type-2 modification.

 

 

Step 5: Add link 4–3, this is Type-4 modification.

 

 

Now, the modified Z-Bus is given by

 

 

Example 3.14

Obtain the Z_Bus of the power system network shown in Figure 3.18(a) by using building algorithm. The self and mutual reactances are indicated in the figure. Take Bus (1) as reference.

Solution:

 

 

Fig 3.18(a) Power System for Example 3.14

 

 

Fig 3.18(b) Graph of the Power System for Example 3.14

 

Consider the graph in Figure 3.18(b).

Step 1: Add branch 1–2, this is a Type-1 modification.

 

 

Step 2: Add branch L₂, this is a Type-2 modification.

 

 

Step 3: Add the link L₁, which is mutually coupled with L₂. This is a Type-4 modification. The primitive impedance and admittance matrices are:

 

 

The augmented impedance matrix is:

 

 

Now, the modified Z_Bus is given by eliminating the fictitious bus,

 

 

Step 4: Add link 1– 3, this is a Type-3 modification.

 

 

Now, the modified Z_Bus is given by eliminating the fictitious bus,

 

 

Example 3.15

In the power system network shown in Figure 3.18(a), the line L2 is removed between buses 2 and 3. Obtain the modified Z_Bus matrix.

Solution:

Removal of network element L₂, which is mutually coupled to L₁ can be obtained as:

 

 

where, the indices a = 2; b = 3; c = 2; d = 3;

The primitive impedance and admittance matrices before elimination are:

 

 

The modified primitive admittance sub-matrix after elimination of element is:

 

 

Before elimination

After elimination of element

 

 

Example 3.16

For the graph of a power system network shown in Figure 3.19. Obtain the Z-Bus matrix by building algorithm. The self reactance of each element is j0.2 p. u. Branch elements are shown as dark lines and link elements as dotted lines. Take mutual reactance as j0.01 p. u.

 

 

Fig 3.19 Graph of the Power System for Example 3.16

 

Solution:

In the figure take the branch between 2 and 3 as L₁ and link as L₂.

Step 1: Add branch 1–2, this is Type-1 modification.

 

 

Step 2: Add branch L₁, this is Type-2 modification.

 

 

Step 3: Add link L₂, which is mutually coupled with L₁. This is a Type-4 modification.

 

 

 

Step 4: Add link 1–3, this is a Type-3 modification.

 

 

Now, the modified Z_Bus is given by

Example 3.17

If one of the lines in Figure 3.18 is removed, modify the Z_Bus matrix.

Solution:

Let the line 1–3 in Figure 3.18 be removed. For this we should add a link across 1 and 3 such that its value is —j0.2 p. u.

 

 

Now add the new link across 1 and 3. This is Type-3 modification.

 

 

The modified Z_Bus is given by

 

Example 3.18

Let the lines L₁ and L₄ be mutually coupled to each other with reactance j0.1 p. u. Obtain Z_Bus using the building algorithm.

Solution:

The graph of the power system is shown in Figure 3.20.

Step 1: Add branch L₁ between 0 and 1, This is Type-1 modification.

 

 

Step 2: Add branch L₂ between 1 and 2, this is a Type-2 modification.

 

 

Step 3: Add link L₃ between 1 and 2, this is a Type-4 modification.

 

 

Now, the modified Z_bus is given by

 

 

Fig 3.20 Graph of the Power System in Example 3.18

 

 

Step 4: Add link L₄ between 0 and 2, which is mutually coupled with L₁. This is a Type-3 modification.

 

 

The primitive impedance matrix is,

 

 

Now, the modified Z_bus is given by

 

 

Example 3.19

In Figure 3.20, if the element L₄ is removed, find the modified Z_Bus.

Solution:

Removal of network element L₄, which is mutually coupled to L₁, is obtained from

 

 

Where the indices a = 0; b = 2; c = 0; d = 1.

The primitive impedance matrix is

 

 

The modified primitive admittance sub-matrix is

 

 

Example 3.20

Obtain the bus impedance matrix of the power system network shown in Figure 3.21(a). Take reactance of each line as j0.3 p. u.

Solution:

 

 

Fig 3.21(a) Power System for Example 3.20

 

 

Fig 3.21(b) Graph of the Power System for Example 3.20

 

The graph for the network is given in Figure 3.21(b)

Step 1: Add branch L₁ between 0 and 1. This is Type-1 modification.

 

 

Step 2: Add branch L₂ between 1 and 2. This is Type-2 modification.

 

 

Add link L₃ between 1and 2. This is Type-4 modification.

 

 

Now, the modified Z_bus is given by eliminating the fictitious bus l

 

 

Step 4: Add link L₄ between 0 and 2. This is Type-3 modification.

 

 

Now, the modified Z-bus is given by eliminating the fictitious bus l

 

 

Example 3.21

If the line L₁ between the buses 0 and 1 is removed from Figure 3.21(a), modify the Z_Bus.

Solution:

If the line L₁ in Figure 3.20 is removed, the modified Z_Bus can be obtained by adding a link in parallel with the element such that the impedance of the link added equals to the negative of the impedance of the element to be removed, as there is no mutual coupling between the elements. This is the removal of a Type-1 modified element.

 

 

By adding the link — j0.3 p. u between 0 and 1, we get

 

 

 

Figure 3.21(c) Graph of the Power System for Example 3.20 with L₁ removed

 

Now, the modified Z-bus is given by:

 

 

Example 3.22

If the line L₂ between the buses 1 and 2 is removed from Figure 3.21(a), modify the Z_Bus.

Solution:

If the line L₂ in the figure is removed, the modified Z _Bus can be obtained by adding a link in parallel with the element such that the impedance of the link added equals to the negative of the impedance of the element to be removed, as there is no mutual coupling between the elements. This is the removal of a Type-2 modified element.

 

 

By adding link —j0.3 p. u. between 1 and 2, we get:

 

 

Now, the modified Z_Bus is given by eliminating the fictitious bus l,

 

Example 3.23

If the line L₃ between the buses 1 and 2 is removed from Figure 3.21(a), modify the Z_Bus.

Solution:

If the line L₃ in the figure is removed, the modified Z-Bus can be obtained by adding a link in parallel with the element such that impedance of the link added equals to the negative of the impedance of the element to be removed, as there is no mutual coupling between the elements. This is the removal of a Type-4 modified element.

 

 

By adding link —j0.3 p. u. between 1 and 2, we get

 

 

Now, the modified Z_Bus is given by eliminating the fictitious bus l

 

Example 3.23

If the line L₄ between the buses 0 and 2 is removed from Figure 3.21(a), modify the Z_Bus.

Solution:

If the line L₄ in the figure is removed, the modified Z_Bus can be obtained by adding a link in parallel with the element such that impedance of the link added equals to the negative of the impedance of the element to be removed, as there is no mutual coupling between the elements. This is the removal of a Type-3 modified element.

 

 

By adding link —j0.3 p. u. between 0 and 2, we get

 

 

Now, the modified Z_Bus is given by eliminating the fictitious bus l

 

Example 3.24

Let the lines L₁ and L₄ be mutually coupled to each other with reactance j0.1 p. u. Obtain the Z_Bus.

Solution:

The graph is given in Figure 3.21(d).

 

 

Fig 3.21(d) Consideration of Mutual Coupling

 

Step 1: Add branch L₁ between 0 and 1. This is Type-1 modification.

 

 

Step 2: Add branch L₂ between 1 and 2. This is Type-2 modification.

 

 

Step 3: Add link L₃ between 1 and 2. This is Type-4 modification. The augmented impedance matrix is:

 

 

where,

 

 

The augmented impedance matrix with fictitious bus l,

 

 

Now, the modified Z_Bus given by eliminating fictitious bus l is:

 

 

Step 4: Add link L₄ between 0 and 2, which is mutually coupled with L₁. This is a Type-3 modification.

 

 

The primitive impedance matrix is,

 

 

Now, the modified Z_Bus is given by eliminating the fictitious bus l

 

 

Example 3.25

In Figure 3.21(a), if the element L₄ is removed, find the Z_Bus from the Z_Bus obtained in Example 3.24.

Solution:

Removal of network element L₄, which is mutually coupled to L₁, is obtained from

 

 

Where the indices are a = 0, b = 2, c = 0, d = 1.

The primitive impedance matrix corresponding to the element to be removed is:

 

 

The modified primitive admittance sub-matrix is,

 

 

where,

 

Questions from Previous Question Papers

1. Give the applications of the Z_Bus building algorithm.
2. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-1 modification. Neglect mutual effect.
3. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-2 modification. Neglect mutual effect.
4. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-3 modification. Neglect mutual effect.
5. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-4 modification. Neglect mutual effect.
6. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-1 modification. Consider the mutual effect.
7. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-2 modification. Consider the mutual effect.
8. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-3 modification. Consider the mutual effect.
9. For a partial n-bus power system Z_Bus is available. An element needs to be added to the existing network. Now describe how the step-by-step procedure shall modify the Z_Bus for Type-4 modification. Consider the mutual effect.
10. Explain how an element can be deleted or how its impedance value can be changed. Neglect Mutual effect.
11. Explain how an element can be deleted or how its impedance value can be changed. Consider the mutual effect of elements.
12. Build the Z_Bus for a power system whose element data is given in the following table.

    Element No.	Connected Between Bus Nos.	Self Reactance in P. U.
    1	1–2	0.3
    2	1–3	0.4
    3	2–3	0.5

13. Modify the Z_Bus that is obtained in Q. 3.12, if a line is added between buses 1 and 3 with a self-reactance value of j0.5 p. u.
14. Modify the Z_Bus that is obtained in Q. 3.12, if a line is added between buses 1 and 3 with a self-reactance value of j0.5 p. u. reactance and a mutual reactance of j0.01 p.u.
15. Explain what do you mean by a partial network.
16. Explain representation of an element in admittance form and impedance form.
17. Describe the algorithm for formation of bus impedance matrix for addition of a branch.
18. Describe the algorithm for formation of bus impedance matrix for addition of a link.
19. Derive an expression for addition of a link to a network with mutual inductance.
20. Derive an expression for adding a branch element between two buses in the Z_bus building algorithm with mutual inductance.
21. Explain the modifications necessary in the Z_Bus when a mutually coupled element is removed or its impedance is changed.
22. What the advantages of Z_Bus building algorithm? Explain what is primitive network, primitive admittance and impedance matrix. Explain by giving an example.
23. Consider the graph of a power system network shown in Figure Q1.

     

    

     

    Fig Q1

     

    Per unit impedance values of various elements are:

    X_a = j1.25; X_b = j0.25; X_c = j0.4; X_d = j1.25; X_e = j0.125; X_f = j0.2

    Determine the bus impedance matrix for the network connecting above impedances.

24. Describe the procedure of modifications in the existing Z_bus by adding mutually coupled branch from existing buses (P) and (K).
25. Obtain the bus impedance matrix by the step-by-step method. A three-bus system having reference node (4), has line impedances in per unit as shown in Figure Q2.

     

    

     

    Fig Q2

     

Competitive Examination Questions

1. The bus numbers and impedances are marked. The bus impedance matrix of this network is

    

   

    

   [IES 1996 Q NO: 106]

2. A power system network consists of three elements 0—1, 1—2 and 2—0 of p.u. impedances 0.2, 0.4 and 0.4 respectively. Its bus impedance matrix is given by

    

   

    

   [IES 1997 Q NO: 42]

3. With the usual notation, impedance matrix for the system shown in the accompanying figure is

    

   
   

    

   [IES 1999 Q NO: 60]

4. The bus admittance matrix of a power system is as shown below. The impedance of the line between buses 2 and 3 will be

    

   

    

   1. j0.1
   2. —j0.1
   3. j0.2
   4. —j0.2

    

   [IES 2003 Q NO: 30]

5. Normally Z_Bus matrix is a
   1. Null matrix
   2. Sparse matrix
   3. Full matrix
   4. Unity matrix

    

   [IES 2004 Q NO: 111]

6. The bus impedance matrix of a 4-bus power system is given by

    

   

    

   A branch having an impedance of j0.2Ω is connected between bus 2 and the reference. Then the values of Z_22,new and Z_23,new of the bus impedance matrix of the modified network are respectively.

   1. j 0.5408 Ω and j 0.4586 Ω
   2. j 0.1260 Ω and j 0.0956 Ω
   3. j 0.5408 Ω and j 0.0956 Ω
   4. j 0.1260 Ω and j 0.1630 Ω

    

   [GATE 2003 Q.No. 11]