G-S algorithm with generator Q limits

The Gauss–Seidel (G-S) method solves the load-flow equations by rewriting each bus equation to give $V_i$ in terms of the other voltages and updating the buses one by one, using the newest values immediately.

Example network

Consider a 3-bus system: bus 1 slack ($V_1 = |V_1|\angle 0$ given), bus 2 PV (generator, $P_2$, $|V_2|$ given, $Q_{2,min} \le Q_2 \le Q_{2,max}$), bus 3 PQ (load, $P_3$, $Q_3$ given).

   G1                      G2
   |  slack       PV bus   |
  (1)----------------------(2)
     \                    /
      \                  /
       ---------(3)------
                 |
               Load (PQ bus)

Basic G-S equations

From $I_i = \sum_k Y_{ik}V_k$ and $P_i - jQ_i = V_i^*I_i$:

$$V_i^{(r+1)} = \frac{1}{Y_{ii}}\left[\frac{P_i - jQ_i}{V_i^{*(r)}} - \sum_{k=1}^{i-1}Y_{ik}V_k^{(r+1)} - \sum_{k=i+1}^{n}Y_{ik}V_k^{(r)}\right]$$ $$Q_i = -\operatorname{Im}\left\{V_i^*\sum_{k=1}^{n}Y_{ik}V_k\right\}$$

Algorithm

1. Read data and form $Y_{bus}$. Net injections $P_i = P_{Gi} - P_{Di}$, $Q_i = Q_{Gi} - Q_{Di}$.
2. Initial values (flat start): slack $V_1$ as given; PV buses $|V_i| = |V_i|^{sp}$, $\delta_i = 0$; PQ buses $V_i = 1.0\angle 0^\circ$. Choose tolerance $\varepsilon$ and acceleration factor $\alpha$ (about 1.4–1.6). Set iteration count $r = 0$.
3. For each bus $i = 2, \ldots, n$ (slack bus skipped):
   • If PQ bus: compute $V_i^{(r+1)}$ from the G-S equation with specified $P_i$, $Q_i$.
   • If PV bus:
     1. Compute $Q_i^{(r+1)} = -\operatorname{Im}\{V_i^{*(r)}\sum_k Y_{ik}V_k\}$ using the latest voltages.
     2. Check Q limits:
        • If $Q_{i,min} \le Q_i^{(r+1)} \le Q_{i,max}$: compute $V_i$ from the G-S equation with this $Q_i$, keep its angle $\delta_i$, and reset its magnitude to $|V_i|^{sp}$: $V_i = |V_i|^{sp}\angle\delta_i$.
        • If $Q_i > Q_{i,max}$: set $Q_i = Q_{i,max}$; if $Q_i < Q_{i,min}$: set $Q_i = Q_{i,min}$. Treat the bus as a PQ bus for this iteration: compute $V_i$ from the G-S equation and do not reset its magnitude.
   • Acceleration (PQ buses): $V_{i,acc}^{(r+1)} = V_i^{(r)} + \alpha\left(V_i^{(r+1)} - V_i^{(r)}\right)$.
   • Use the new $V_i$ immediately for the following buses.
4. Convergence check: $\Delta V_{max} = \max_i |V_i^{(r+1)} - V_i^{(r)}|$. If $\Delta V_{max} > \varepsilon$, set $r = r + 1$ and repeat step 3. (A bus that was switched to PQ may return to PV if its voltage later allows Q to come back within limits.)
5. After convergence:
   • Slack bus power: $P_1 - jQ_1 = V_1^*\sum_k Y_{1k}V_k$.
   • Q at PV buses.
   • Line flows: $S_{ik} = V_i\left[(V_i - V_k)y_{ik} + V_i\frac{y'_{ik}}{2}\right]^*$.
   • Losses: $S_{L,ik} = S_{ik} + S_{ki}$; total loss $= \sum S_{L}$.
6. Print results.

Flow chart (summary)

 Form Ybus -> flat start -> r = 0
        |
        v
  for i = 2..n:
    PQ? -> new Vi (G-S), accelerate
    PV? -> find Qi -> within limits?
           yes: new Vi, reset |Vi|
           no : Qi = limit, treat as PQ
        |
        v
  max|dV| < eps ? --no--> r = r+1, repeat
        | yes
        v
  slack power, line flows, losses