Introduction

The Y-Bus matrix, also known as nodal admittance matrix, is a key component in power flow analysis, providing a mathematical representation of the power systems. It is a sparse and symmetric matrix that characterizes the admittance between all pairs of connected buses in a power system. This matrix is used in solving power flow equations to determine the steady-state operating conditions of the power systems, making it a backbone for their analysis and design.

Y-Bus Matrix Formation -

To obtain Y-Bus matrix for the interconnected power system, consider a simple network shown in Figure 1.

Figure 1. Sample network for 5-bus power system.

To obtain Y-Bus matrix for the power system, shown in Figure 1. For the Y-Bus matrix formation, all the transmission lines are represented by the π-model (please follow any of power system textbook for more details). For the current injection at each node (bus), the node-voltage equations need to be considered which are based on Kirchoff’s current law (KCL). The impedances of the transmission lines are converted to admittances as per the equation (1) given below.

y_ij = 1/ z_ij = 1/ (r_ij+ x_ij) ..............................(1)

The admittance diagram for the 5-bus power system is shown in Figure 2, where series line admittance (y_ij) and half line charging susceptance (b_ij) are shown. The admittance of the transformers 1 and 2 is represented by y_T1 and y_T2, respectively. As the transformer connected between buses 1 and 2 has a tap ratio of 1:1, its series admittance is equal to the admittance of the transformer, i.e., y_T1 and the line charging susceptances equal to zero. Whereas the transformer connected between the buses 3 and 5 having a tap ratio of 1:t, the series admittance is equal to ' t ' times of the admittance of the transformer, i.e., ty_T2, and line charging susceptances are equal to t(1-t)y_T2 and (1-t)y_T2 connected at buses 3 and 5, respectively. Now, applying KCL to the all buses, the node-voltage equations are given below.

Bus 1: I₁ = y_T1(V₁ - V₂)

Bus 2: I₂ = y_T1(V₂ - V₁) + y₂₃(V₂ - V₃) + y₂₄(V₂ - V₄) + b₂₃V₂ + b₂₄V₂

Bus 3: I₃ = y₂₃(V₃ - V₂) + y₃₄(V₃ - V₄) + ty_T2(V₃ - V₅) + b₂₃V₃ + t(t-1)y_T2V₃ + b₃₄V₃

Bus 4: I₄ = y₂₄(V₄ - V₂) + y₃₄(V₄ - V₃) + (b₂₄ + b₃₄)V₄

Bus 5: I₅ = ty_T2(V₅ - V₃) + (1-t)y_T2V₅

Rearranging the terms, we get

Bus 1: I₁ = y_T1V₁ - y_T1V₂

Bus 2: I₂ = -y_T1V₁ + (y_T1 + b₂₃ + y₂₃ + b₂₄ + y₂₄)V₂ - y₂₃V₃ - y₂₄V₄

Bus 3: I₃ = -y₂₃V₂ + (y₂₃ + b₂₃ + ty_T2 + t(t-1)y_T2 + y₃₄ + b₃₄) V₃ - y₃₄V₄ - ty_T2V₅

Bus 4: I₄ = -y₂₄V₂ - y₃₄V₃ - (y₂₄ + b₂₄ + y₃₄ + b₃₄)V₄

Bus 5: I₅ = -ty_T2V₃ + (ty_T2 + (1-t)y_T2)V₅

Figure 2. Equivalent admittance diagram of the 5-bus power system.

Writing them in the matrix form as –

Where-

Y₁₁=y_T1;   Y₁₂=Y₂₁=-y_T1;   Y₁₃=Y₁₄=Y₁₅=0
Y₂₂=y_T1+b₂₃+y₂₃+b₂₄+y₂₄;   Y₂₃=-y₂₃;   Y₂₄=-y₂₄;   Y₂₅=0;
Y₃₃=y₂₃+b₂₃+ty_T2+t(t-1)y_T2+y₃₄+b₃₄;   Y₃₂=-y₃₂;   Y₃₄=-y₃₄;
Y₃₅=Y₅₃=-ty_T2;   Y₄₁=0;   Y₄₂=-y₂₄;   Y₄₃=-y₃₄;
Y₄₄=y₂₄+b₂₄+y₃₄+b₃₄;   Y₄₅=Y₅₄=0;   Y₅₅=ty_T2+(1-t)y_T2;

Similar to this, we can generalize the Y-Bus matrix for the n-bus power system as given below.

Or,

I_bus = Y_bus.V_bus

Where, I_bus is the vector of the injected bus currents and V_bus is the vector of bus voltages measured from the reference node. Y_bus is known as the bus admittance matrix or or nodal admittance matrix.

With the above explanation, the Y_bus matrix for the power system can be formulated based on the following two conditions.

1. For the lines, without transformer:

• The diagonal element of each node is the sum of admittances connected to it. It can be calculated as:

Y_ii = n ∑ j=1 y_ij + b_ij   

• The non-diagonal elements for each node are the negative of the series admittance of the connected line. It can be calculated as:

Y_ij = Y_ji = -y_ij ,   ∀ j ≠ i

2. For the lines, with transformer:

• The diagonal element of each node is the sum of admittances connected to it. The diagonal elements for primary and secondary side of the transformer are calculated as:

For primary side bus,

Y_ii = n ∑ j=1 y_ij + ty_T + t(t-1)y_T + b_ij   

For secondary side bus,

Y_jj = n ∑ j=1 y_ij + ty_T + (1-t)y_T + b_ij   

• The non-diagonal elements are calculated by multiplying transformer turn ratio with the negative of the series admittance of the connected line. It can be calculated as –

Y_ij = Y_ji = -ty_ij ,   ∀ j ≠ i

The Y-Bus matrix is symmetric along the leading diagonal, due to this only upper triangular nodal admittance matrix is required improving the computational performance. Also, the Y-Bus matrix is a sparse matrix due to the interconnection of each bus with only a few nearby buses.