Need for load flow; NR and decoupled equations

A load flow (power flow) study finds the steady-state voltage magnitude and angle at every bus of a network for a given load and generation, and from these the real and reactive power flows and losses in every line.

When and why it is needed

• Planning and expansion: to check that a proposed line, transformer or generator keeps all bus voltages within limits (about ±5%) and no line is overloaded.
• Daily operation: to schedule generation, set transformer taps and switch capacitors/reactors for the forecast load.
• Contingency studies: to see what happens if a line or generator trips (N−1 security).
• Input to other studies: economic dispatch, loss calculation, and the pre-fault voltages needed for fault and stability studies.

Load flow equations for the N-R method

With $V_i = |V_i|\angle\delta_i$ and $Y_{ij} = |Y_{ij}|\angle\theta_{ij}$:

$$\begin{aligned} P_i &= \sum_{j=1}^{n}|V_i||V_j||Y_{ij}|\cos(\theta_{ij} - \delta_i + \delta_j) \\ Q_i &= -\sum_{j=1}^{n}|V_i||V_j||Y_{ij}|\sin(\theta_{ij} - \delta_i + \delta_j) \end{aligned}$$

Linearising about the present estimate gives

$$\begin{bmatrix} \Delta P \\ \Delta Q \end{bmatrix} = \begin{bmatrix} J_1 & J_2 \\ J_3 & J_4 \end{bmatrix} \begin{bmatrix} \Delta\delta \\ \Delta|V| \end{bmatrix}$$

where $J_1 = \partial P/\partial\delta$, $J_2 = \partial P/\partial|V|$, $J_3 = \partial Q/\partial\delta$, $J_4 = \partial Q/\partial|V|$. The full Jacobian is rebuilt and solved every iteration.

Assumptions for decoupled load flow

1. Line resistance is much smaller than reactance ($R \ll X$), so $\theta_{ij} \approx 90^\circ$.
2. Angle differences across lines are small, so $\cos(\delta_i - \delta_j) \approx 1$ and $\sin(\delta_i - \delta_j) \approx 0$.
3. Hence $P$ depends mainly on $\delta$ and $Q$ mainly on $|V|$, so $J_2 \approx 0$ and $J_3 \approx 0$:

$$\begin{aligned} \Delta P &= J_1\,\Delta\delta \\ \Delta Q &= J_4\,\Delta|V| \end{aligned}$$

4. For the fast decoupled method, also take $Q_i \ll B_{ii}|V_i|^2$ and $|V_i| \approx 1$, which gives $\dfrac{\Delta P}{|V|} = -B'\,\Delta\delta$ and $\dfrac{\Delta Q}{|V|} = -B''\,\Delta|V|$, where $B'$ and $B''$ are constant matrices built from $B_{bus}$.

Computational advantages over N-R

• Two small matrices replace one large Jacobian, so memory is roughly halved.
• In the fast decoupled form, $B'$ and $B''$ are constant and are factorised only once.
• Each iteration is much faster. It needs a few more iterations, but total time is lower, which suits on-line and contingency studies.