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Load Flow Analysis

Bus voltages · branch currents · radial vs looped · software tools

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.

Output	What you do with it	Section reference
Bus voltages (magnitude + angle)	Verify each load receives within tolerance (±5% per ANSI C84.1)	§01, §06
Branch currents	Confirm wires not overloaded; check transformer loading	§06, §09
Real + reactive power at each node	Identify where reactive power is generated/consumed; PFC placement	§15
Transformer tap recommendations	Adjust no-load taps to optimize voltage profile	§09
Generator dispatch (if multiple sources)	Determine which gen carries which load	§19
Loss analysis	Find inefficient feeders; size correction	—

Radial vs Looped vs Networked

Topology	Description	Hand calc?	Typical use
Radial	Single source feeding tree of loads. No closed loops.	Yes — work from source to ends	99% of commercial / industrial buildings, residential
Looped	Two sources or feeders meet, with a normally-open tie	Possible but tedious — two cases (each tie position)	Critical commercial (hospitals, data centers); urban distribution
Networked	Multiple sources, multiple paths, any load can be supplied through several routes	Software only — Newton-Raphson or similar iterative solver	Utility transmission, downtown urban (network protector grids)

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.

₂? L2 100 kVA PF=0.9 F2: 4/0, 100ft R = 0.012 Ω Bus 3 — V₃? L3 75 kVA PF=0.95

Three-bus radial example — solve for V at Bus 2 and Bus 3

Solution

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
Current in F1 = sum of downstream loads.
I_F1 = I₂ + I₃ = 120.3 + 90.2 = 210.5 A (assuming similar PFs)
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) ✓
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) ✓
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:

Software	Use case	Note
SKM PowerTools	Industry standard for industrial / commercial	Steep learning curve. Comprehensive.
ETAP	Industrial focus. Strong for arc flash + protection coordination integration	Most popular for power plant + petrochemical work
EasyPower	User-friendly. Good for new engineers.	Excellent integration of load flow + arc flash + coordination
PowerWorld	Transmission system focus	Used by utilities + ISOs
PSS/E or PSCAD	Utility / transmission planning + transient analysis	Specialized

Worked Example 1 — Atlas DC1 Voltage Profile

Example 01 · Atlas DC1 spineVoltage at every bus from utility through to rack PDU

Bus	Voltage	Drop from upstream	%VD running total
Utility 12.47 kV (PCC)	12.47 kV	—	0%
TX-A primary	12.47 kV	negligible (short MV cable)	0%
TX-A secondary (480V SWGR-A)	478 V	5.75% × loading × cos(impedance angle) = ~0.4%	~0.4%
UPS-A1 input	477 V	0.6% (250 ft feeder)	~1.0%
UPS-A1 output (regulated)	480 V	UPS regulates to setpoint — eliminates upstream variation	0% (re-referenced)
PDU-A1 input (480V)	477 V	0.6% (250 ft from UPS)	~0.6%
PDU-A1 output (415V at xfmr secondary)	413 V	3.5% × loading at PDU xfmr ~ 0.5%	~1.1% from UPS
RPP-A1-1 (415V)	411 V	0.6% (50 ft from PDU)	~1.7%
Rack PDU strip (240V phase-neutral)	237 V	0.4% (10 ft branch)	~2.1%

UPS as a voltage reset

The UPS is more than backup power — it actively regulates output voltage regardless of input. Upstream voltage variations (utility sag, transformer drops) don't propagate downstream. Total VD from UPS to rack: ~2.1%, well within IT equipment tolerance (±10% typical).

Worked Example 2 — Apartment Building VD Across Service

Example 02 · Alternate scale50-unit apartment · 980 A demand · 80 ft service feeder · check VD at panel

Service: 1200 A breaker, 3 sets of 750 kcmil Cu, 80 ft.
VD on service feeder (from §06): 0.95 V at 980 A. = 0.46% VD
VD on per-unit feeder (200 A panel, 50 ft, #2/0 Cu):
VD = 2 × 100 (1φ) × 0.13 × 50 / 1000 = 1.3 V on 240V = 0.54%
VD on branch circuit (worst — 30A range, 50 ft):
VD = 2 × 30 × 0.78 (#8) × 50 / 1000 = 2.34 V on 240V = 0.98%
Total worst-case VD (utility transformer secondary to range outlet): 0.46 + 0.54 + 0.98 = ~2.0%. Well within 5% NEC recommendation.

Drill — Quick Self-Check

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

Drill 1 · VD on feeder

200 ft of #2 Cu (R = 0.20 Ω/kft), 100 A at 480V 3φ. %VD?

Drill 2 · Radial vs looped

Hand-calc possible for which topology?

Drill 3 · Voltage profile

Atlas DC1 utility 12.47kV → server. How many transformations?

Drill 4 · PCC

What is the Point of Common Coupling?

Drill 5 · UPS as voltage reset

Voltage at UPS output regardless of input variation?

Voltage Regulation — The Theory

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

	Voltage Drop (%VD)	Voltage Regulation (%VR)
Definition	(V_source − V_load) / V_nominal × 100	(V_no-load − V_full-load) / V_full-load × 100
Reference	Nominal voltage	Load-side full-load voltage
Used for	Conductor sizing	Transformer + generator performance
Typical limit	≤ 3% feeder, ≤ 5% combined (NEC 215.2 IN)	≤ 3-5% for most transformers (depends on application)

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:

Real power transferred (lossless line approximation)

P = (|V_S| × |V_R| / X) × sin δ

δ = phase angle between sending and receiving voltages. Maximum theoretical transfer at δ = 90° (steady-state stability limit).

Reactive power received

Q_R = (|V_R|/X) × (|V_S| × cos δ − |V_R|)

Reactive power flow depends on voltage MAGNITUDE difference. Real power flow depends on voltage ANGLE difference. This is the fundamental decoupling that makes power flow analysis tractable.

The PV-PQ decoupling — why power flow is solvable

Real power transfer is governed by voltage ANGLE differences (δ). Reactive power transfer is governed by voltage MAGNITUDE differences (|V|). For typical lines (X >> R), they're nearly independent. This decoupling is why we can solve P and Q separately in iterative load flow algorithms (Newton-Raphson, fast-decoupled).

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.

If You See THIS, Think THAT

If you see…	Think / use…
"Load flow analysis"	Steady-state V, I, P, Q at every node. Software for complex; hand calc for radial.
"Voltage profile"	V vs distance plot. Tells you where to add tap adjustments or upsize feeders.
"Voltage regulation"	(V_NL − V_FL) / V_FL × 100. NEC informational note ≤ 5% total.
"Reactive power flow" / VARs	Inductive loads sink VARs; capacitors source them. Flows from generator/cap → load.
"Looped" or "networked" system	Software required. Hand calc impractical.
"Radial system"	Hand calc OK. Work source-to-load.
"PCC" (Point of Common Coupling)	Boundary between user + utility. IEEE 519 limits apply here.
"SKM/ETAP/EasyPower"	Power system software. SKM = legacy industrial; ETAP = industrial focus; EasyPower = user-friendly.
UPS in the system	Voltage reset point. Upstream variation doesn't propagate downstream.