For a 15-bus system power system with 3 voltage-controlled buses, the size of Jacobian matrix of Newton-Raphson method used to solve load flow problem is:

Question

Prepp Team · Accepted Answer

Understanding Load Flow with Newton-Raphson

Load flow or power flow studies are essential analyses performed in power systems to determine the voltage magnitudes and angles at all buses, as well as active and reactive power flows through transmission lines and transformers, and system losses under steady-state operating conditions.

The load flow problem involves solving a set of non-linear algebraic equations that describe the power balance at each bus in the system. The Newton-Raphson method is a widely used iterative technique to solve these non-linear equations due to its fast convergence characteristics.

Bus Types in Power Systems Load Flow

For load flow analysis, buses in a power system are classified into three main types based on the quantities that are known and unknown:

Slack Bus (or Reference Bus): This bus is typically chosen as the reference bus. Its voltage magnitude $(|V|)$ and voltage angle $(\delta)$ are known (usually $|V|=1.0$ p.u. and $\delta=0$ degrees). This bus handles the power mismatch (difference between total generation and total load plus losses) for the entire system. Both active power (P) and reactive power (Q) are unknown at the slack bus.
PV Bus (Voltage-Controlled Bus): For these buses, the active power (P) injected (or drawn) and the voltage magnitude $(|V|)$ are known (usually generators are represented as PV buses as they control voltage). The unknown quantities to be determined are the voltage angle $(\delta)$ and the reactive power (Q).
PQ Bus (Load Bus): These represent buses where the active power (P) and reactive power (Q) are known (usually loads are represented as PQ buses). The unknown quantities to be determined are the voltage magnitude $(|V|)$ and the voltage angle $(\delta)$.

Newton-Raphson Method and Jacobian Matrix Size

The Newton-Raphson method solves the load flow equations iteratively by linearizing them around an initial guess. This linearization results in a system of linear equations that relates the change in unknown variables to the mismatch in known quantities (active and reactive power). This system is expressed as:

\Delta \mathbf{f} = \mathbf{J} \Delta \mathbf{x}

where:

$\Delta \mathbf{f}$ is the vector of power mismatches ($\Delta P$ for PV and PQ buses, and $\Delta Q$ for PQ buses).
$\Delta \mathbf{x}$ is the vector of corrections to the unknown state variables ($\Delta \delta$ for PV and PQ buses, and $\Delta |V|$ for PQ buses).
$\mathbf{J}$ is the Jacobian matrix, which contains the partial derivatives of the power mismatch equations with respect to the unknown state variables.

The size of the Jacobian matrix is determined by the number of equations being solved and the number of unknown variables being updated. In the Newton-Raphson load flow, the number of equations equals the number of unknown variables.

Calculating the Jacobian Matrix Size for the 15-Bus System

Given the power system:

Total number of buses, $N = 15$.
Number of voltage-controlled buses (PV buses), $N_{PV} = 3$.

We assume a standard load flow setup which includes one slack bus.

Number of Slack buses, $N_{Slack} = 1$.

The remaining buses must be PQ buses.

Number of PQ buses, $N_{PQ} = N - N_{Slack} - N_{PV} = 15 - 1 - 3 = 11$.

Now, let's identify the unknown variables that the Newton-Raphson method solves for:

Voltage angles ($\delta$) are unknown for all buses except the slack bus. The number of buses with unknown $\delta$ is $N - N_{Slack} = 15 - 1 = 14$. These are the $N_{PV}$ buses and the $N_{PQ}$ buses ($3 + 11 = 14$).
Voltage magnitudes $(|V|)$ are unknown only for the PQ buses. The number of buses with unknown $|V|$ is $N_{PQ} = 11$.

The total number of unknown state variables is the sum of the unknown voltage angles and unknown voltage magnitudes:

Total unknowns = (Number of unknown $\delta$) + (Number of unknown $|V|$) = $14 + 11 = 25$.

The number of equations in the Newton-Raphson method corresponds to the power mismatches that are driven to zero:

Active power mismatch ($\Delta P$) equations are formulated for all buses except the slack bus ($N - N_{Slack}$). Number of $\Delta P$ equations = $14$. (These correspond to the PV and PQ buses).
Reactive power mismatch ($\Delta Q$) equations are formulated for the PQ buses ($N_{PQ}$). Number of $\Delta Q$ equations = $11$. (For PV buses, $|V|$ is controlled, not Q, so $\Delta Q$ equations are not used to update $|V|$ but rather Q is calculated after convergence).

The total number of equations is the sum of the $\Delta P$ and $\Delta Q$ equations:

Total equations = (Number of $\Delta P$ equations) + (Number of $\Delta Q$ equations) = $14 + 11 = 25$.

The size of the Jacobian matrix is (Total number of equations) $\times$ (Total number of unknowns).

Jacobian Matrix Size = $25 \times 25$.

This size corresponds to the dimensions of the matrix $\mathbf{J}$ in the equation $\Delta \mathbf{f} = \mathbf{J} \Delta \mathbf{x}$. The number of rows in $\mathbf{J}$ is the number of equations (25), and the number of columns is the number of unknown variables (25).

Here is a summary:

Bus Type	Count	Unknown Variables	Number of $\Delta P$ Equations	Number of $\Delta Q$ Equations
Slack	1	-	-	-
PV	3	$\delta$, Q	1 (for $\Delta P$)	-
PQ	11	$\|V\|$, $\delta$	1 (for $\Delta P$)	1 (for $\Delta Q$)

Total unknown $\delta$: $3 + 11 = 14$
Total unknown $|V|$: $11$
Total unknowns = $14 + 11 = 25$

Total $\Delta P$ equations: $3 + 11 = 14$
Total $\Delta Q$ equations: $11$
Total equations = $14 + 11 = 25$

The size of the Jacobian matrix is therefore $25 \times 25$.

Revision Table: Key Load Flow Concepts

Concept	Description	Calculation Relevance
Total Buses (N)	Total number of nodes in the system.	Base for calculating bus type counts.
Slack Bus	Reference bus, known $\|V\|$ and $\delta$.	Reduces unknown $\delta$ count by 1. No mismatch equations.
PV Bus	Known P and $\|V\|$, unknown $\delta$ and Q.	Contributes 1 unknown $\delta$ and 1 $\Delta P$ equation per bus.
PQ Bus	Known P and Q, unknown $\|V\|$ and $\delta$.	Contributes 1 unknown $\delta$, 1 unknown $\|V\|$, 1 $\Delta P$ equation, and 1 $\Delta Q$ equation per bus.
Jacobian Matrix	Core of Newton-Raphson method.	Size = (Total Equations) $\times$ (Total Unknowns)

Additional Information: Structure of the Jacobian Matrix

The Jacobian matrix $\mathbf{J}$ in the Newton-Raphson load flow is typically partitioned into four submatrices based on the partial derivatives:

$\mathbf{J} = \begin{bmatrix} \frac{\partial P}{\partial \delta} & \frac{\partial P}{\partial |V|} \\ \frac{\partial Q}{\partial \delta} & \frac{\partial Q}{\partial |V|} \end{bmatrix} = \begin{bmatrix} \mathbf{J}_1 & \mathbf{J}_2 \\ \mathbf{J}_3 & \mathbf{J}_4 \end{bmatrix}$

The state variable and mismatch vectors are ordered based on bus types:

$\Delta \mathbf{f} = \begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix}$, $\Delta \mathbf{x} = \begin{bmatrix} \Delta \delta \\ \Delta |V| \end{bmatrix}$

Specifically, the equations $\Delta P$ are for PV and PQ buses, $\Delta Q$ are for PQ buses. The variables $\Delta \delta$ are for PV and PQ buses, $\Delta |V|$ are for PQ buses.

The dimensions of the submatrices are:

$\mathbf{J}_1$: $(N - N_{Slack}) \times (N - N_{Slack})$ (derivatives of $\Delta P$ w.r.t $\Delta \delta$)
$\mathbf{J}_2$: $(N - N_{Slack}) \times N_{PQ}$ (derivatives of $\Delta P$ w.r.t $\Delta |V|$)
$\mathbf{J}_3$: $N_{PQ} \times (N - N_{Slack})$ (derivatives of $\Delta Q$ w.r.t $\Delta \delta$)
$\mathbf{J}_4$: $N_{PQ} \times N_{PQ}$ (derivatives of $\Delta Q$ w.r.t $\Delta |V|$)

For this problem: $N=15$, $N_{Slack}=1$, $N_{PV}=3$, $N_{PQ}=11$. $N - N_{Slack} = 14$.

$\mathbf{J}_1$: $14 \times 14$
$\mathbf{J}_2$: $14 \times 11$
$\mathbf{J}_3$: $11 \times 14$
$\mathbf{J}_4$: $11 \times 11$

The total Jacobian matrix size is the sum of the dimensions: $(14+11) \times (14+11) = 25 \times 25$. This confirms the calculation based on total unknowns/equations.