Load Flow Analysis

Load flow tells you the voltage at every bus, the current in every feeder, and where power factor sags. Done by hand for small systems; done with software (SKM, ETAP, EasyPower) for everything else.

What Load Flow Analysis Computes

Load flow (a.k.a. power flow) is the steady-state solution of the power system: voltage at every bus, current in every feeder, real and reactive power flow at every branch. Without it, you're guessing at voltage drop and PF on complex systems.

Radial vs Looped vs Networked

Hand Calc — 3-Bus Radial

For radial systems, work from source to ends. At each bus, sum the downstream loads, apply the upstream impedance, calculate voltage drop, repeat.

Solution

1. Currents at each bus.

   I₂ (load 2) = 100 × 1000 / (√3 × 480) = 120.3 A at PF 0.9 lag
   I₃ (load 3) = 75 × 1000 / (√3 × 480) = 90.2 A at PF 0.95 lag
2. Current in F1 = sum of downstream loads.

   I_F1 = I₂ + I₃ = 120.3 + 90.2 = 210.5 A (assuming similar PFs)
3. Voltage drop on F1.

   VD_F1 = √3 × I × R = √3 × 210.5 × 0.011 = 4.0 V → V₂ = 480 − 4 = 476 V (0.83% drop) ✓
4. Current in F2 = I₃ only.

   VD_F2 = √3 × 90.2 × 0.012 = 1.9 V → V₃ = 476 − 1.9 = 474 V (1.25% total drop) ✓
5. Total voltage drop budget: ≤ 5%. Atlas system has plenty of margin.

When You Need Software

Hand calc works for small radial. For real systems, use load flow software:

Worked Example 1 — Atlas DC1 Voltage Profile

Worked Example 2 — Apartment Building VD Across Service

Drill — Quick Self-Check

Work each problem mentally; reveal to check. Goal: reflex, not deliberation.

Voltage Regulation — The Theory

Voltage drop and voltage regulation sound similar but mean different things in formal practice.

Two-Bus Power Flow Equations

For a transmission line connecting two buses (sending S, receiving R) with line impedance Z = R + jX and angle δ between bus voltages:

Surge Impedance Loading (SIL) — for transmission

When real power transfer = SIL = V_line² / Z_c (where Z_c is line surge impedance), the line has zero net reactive power along its length. Above SIL → line absorbs reactive (looks inductive). Below SIL → line delivers reactive (looks capacitive). This concept governs reactive compensation strategy on transmission systems.

Steady-State Stability Limit

From the two-bus equation, P_max = |V_S||V_R|/X at δ = 90°. Beyond 90°, the system becomes unstable (small perturbation leads to larger perturbation). Practical operating limits keep δ < 35-40° for adequate stability margin.

Transient Stability — Swing Equation

Steady-state stability tells you whether a small perturbation grows. Transient stability asks the harder question: after a sudden disturbance — a fault and its clearing — will the rotor of a synchronous machine swing back into step with the rest of the grid, or will it slip and require resynchronization (or trip)? The PE Power exam tests three flavors of stability and one fundamental equation.

The swing equation is just Newton's second law applied to the rotor: net torque accelerates angular position. When P_m > P_e (faulted feeder reduces electrical output but turbine keeps pushing), the rotor accelerates and δ rises. When P_e > P_m after fault clearing, the rotor decelerates back. Whether it makes it back without slipping is what the equal-area criterion answers.

Equal-Area Criterion

Plot P_e vs δ (a sin curve from the two-bus equation). On the same axes, draw a horizontal line at P_m. The area above the P_m line and under the post-fault P_e curve, between the clearing angle and the maximum allowed angle, is the decelerating area. The area below the P_m line and above the during-fault P_e curve, from the pre-fault angle to the clearing angle, is the accelerating area.

The practical lever is fault clearing time. Faster clearing → smaller accelerating area → bigger decelerating margin. The minimum-margin clearing time is the critical clearing time (CCT), often 4–10 cycles (67–167 ms) on transmission, much faster on generator buses. Modern transmission protection clears 3-phase faults in 3–6 cycles precisely to stay inside CCT.

Voltage Stability — P-V and Q-V Curves

For a load bus fed through impedance Z from a stiff source, plot V_load against P_load at fixed PF — the result is the "nose curve." As load grows, voltage falls; at some critical load (the nose), an additional ΔP causes V to collapse rather than fall gradually. The distance between current operating point and the nose is the voltage stability margin. Q-V curves are dual: they show how much reactive support is needed to hold a target voltage at a given P. Wind farm ride-through and large IBR plants are sized partly against these curves.

Frequency Stability — Inertia and Governor Response

When generation suddenly exceeds load (loss of a big load) or load exceeds generation (loss of a big plant), system inertia momentarily absorbs the imbalance — frequency rises or falls at the rate-of-change-of-frequency (ROCOF) df/dt = ΔP / (2H · S_system). Governor response then catches it. UFLS (under-frequency load shedding) is the last line of defense: at 59.5 / 59.0 / 58.5 Hz successive blocks of load drop. Inverter-based resources contribute zero physical inertia by default — grid-forming firmware can synthesize it (see §40).

If You See THIS, Think THAT