Power Flow Simulator v1.0

1. Drag a tap slider in the right pane to see real and reactive flows redistribute in real time.
2. Click "Bus voltage heatmap" in the radio toggle below the diagram to see where the system is under- or over-voltage.
3. Right-click a branch or element in the diagram to toggle it in/out of service (simulates an outage).
4. Click any element in the diagram (a bus, a transformer, a load triangle) to select it — dotted lines connect it to its data labels.
5. Right-click a bus to change its type (slack / PV / PQ).
6. Use the toolbar above the diagram to add buses, generators, lines, transformers, loads, and shunt devices.
7. Click "Share link" to generate a URL containing the entire model — paste it anywhere, anyone clicking it sees your exact case.
8. Click "Export animation" with format=Video to record an MP4 of the flow animation, watermarked and ready to share.

Click anywhere in the diagram first to focus it (you'll see a soft blue outline). Then these keys are active:

These shortcuts only fire when the SVG diagram has focus (click it) and when you're not typing in a text input or table cell.

The toolbar above the diagram provides these tool modes (click to activate):

Model management buttons (right side of toolbar)

• New model — clears everything and starts with a blank canvas.
• 8 Bus Demo — loads a small system (slack + PV gen, three transformers, parallel transmission, mixed loads).
• 40 Bus Demo — loads a three-region grid at 345/138/13.8 kV with multiple generators, a PST, and distributed reactive support.
• Save JSON — downloads the model as a .json file to your computer.
• Load JSON — opens a previously saved .json model file.
• Share link — encodes the entire model into a URL hash; anyone clicking the URL sees your exact case.
• Reset labels — clears all manual label position overrides; auto-placement resumes.

View controls

• + / − buttons — zoom in/out (same as keyboard +/− or mouse wheel).
• Fit — auto-frame all content including labels.
• Reset — return to the default zoom and position.
• ⛶ Fullscreen — expand the diagram to fill the entire screen (press Esc to exit).

Buses (the rectangular shapes)

A bus represents a node in the network — physically, a substation busbar or any point where multiple connections meet. Each bus has one of three types:

• Slack bus (one per system, fixed voltage 1.0–1.05 pu, fixed angle 0°): the system's angle reference. Its real and reactive power are computed by the solver to balance the system.
• PV bus (voltage-controlled generator): you specify P and V; the solver computes Q (the reactive support the generator must provide to hold V) and θ (the angle at which it operates).
• PQ bus (load bus, also used for PQ-mode generators): you specify P and Q; the solver computes V and θ.

Right-click any bus to change its type. In voltage-heatmap mode, the bus rectangle's fill color shows its voltage on a blue→white→red scale (low→nominal→high). In angle-heatmap mode, purple→white→orange shows lagging→reference→leading angle.

Branches (the lines between buses)

Branches come in two kinds:

• Lines (plain straight line): a transmission or distribution line, modeled as a π-equivalent with series resistance r, series reactance x, and total shunt charging susceptance b — all in per-unit on the system base of 100 MVA.
• Transformers (two circles connected by a small line): a two-winding transformer with off-nominal turns ratio (tap). Click a transformer to select it, then press R to rotate which side is primary.

Generators (the "G" circles)

A generator symbol attaches to a bus that has hasGenerator: true. The label below the G shows actual P and Q output (computed for slack/PV, fixed for PQ).

Loads (the orange triangles)

A load triangle shows constant-power demand at a bus. Negative load values represent injection (e.g., a distributed generation site).

Shunt compensation (green capacitor symbols, brown reactor symbols)

Shunt caps inject reactive power; shunt reactors absorb it. By default they're modeled as constant-MVAR devices, but you can switch to constant-susceptance (V²B) in the Modeling options group — see "Shunt modeling" below.

Animation triangles

The colored triangles flowing along each branch show the direction and magnitude of power flow. Blue triangles on the upper side of each branch are real power (P, in MW); orange triangles on the lower side are reactive power (Q, in MVAR). Triangle direction encodes sign of flow; both density and speed scale with √(magnitude), so the visual flux (triangles passing a point per second) is proportional to the actual flow magnitude.

The central pedagogical point: P flows from higher angle to lower angle; Q flows from higher voltage to lower voltage. Watch what happens when you adjust a tap or a generator V setpoint — the orange triangles change direction visibly while the blue ones stay roughly the same.

Each row is one generator. Cells gray out when they don't apply to that generator's bus type.

Cells with a violated limit get a pink background. The bus results table shows an asterisk on the V cell when the actual voltage drifted from the setpoint due to a Q-limit hit; hover the asterisk for details.

When Zgen is non-zero, the V column in the bus results table shows the actual terminal voltage at that bus (what loads see). The V setpoint slider in the Controls panel sets the internal EMF — the terminal voltage will drop below this as the gen sources real or reactive power. With Zgen = 0 (default), terminal V equals the setpoint exactly.

Lists every bus that has nonzero base load. Each row gives:

• P load (MW) and Q load (MVAR) — the rated demand at that bus. Negative values represent injection. For the constant-impedance model, this is the value drawn at exactly 1.0 pu; the actual draw scales with V².
• Model — pick how the load behaves with voltage:
  • const P (default) — the load draws fixed P + jQ regardless of bus voltage. This is the standard textbook assumption. Most data-tabulated loads (industrial, motor-dominant, modern electronics with switched-mode supplies) approximate this.
  • const Z — the load looks like a constant impedance, so P and Q both scale with V². At 0.95 pu, the load draws only 0.9025 of rated. Resistive heating, incandescent lighting, and many distribution-level loads approximate this. Mathematically: P_drawn = P_rated · V², Q_drawn = Q_rated · V².
• In service — uncheck to remove this load from the solve (e.g., to simulate a feeder trip). The load doesn't get deleted; just zeroed for the next solve.

The load model affects how the system responds to voltage swings. With constant-power loads, a voltage-collapse scenario can spiral: low V → fixed power demand → more current → more I²X drop → even lower V. With constant-impedance loads, low voltage automatically reduces the demand, providing natural stabilization. Real grids fall somewhere between (the IEEE "ZIP" model splits load into constant Z, constant I, and constant P fractions).

To add a new load to a bus that doesn't have one yet, edit the bus's row in the Bus settings table further down — set Pl/Ql to nonzero values and the load row will appear here automatically.

The "Total load (×)" slider in the Controls panel scales every load uniformly without modifying the underlying values. Useful for scenario sweeps.

When the load model is constant-impedance, the values shown in the P load and Q load columns are the rated values (at 1.0 pu). The actual P and Q drawn at the converged voltage appear in the Bus results table further down.

One row per bus with nonzero Qcap or Qreact:

• Q cap (MVAR) — capacitive injection at the bus, rated at 1.0 pu. Always entered as a positive number.
• Cap model — pick how the capacitor behaves with voltage:
  • const MVAR (default) — the cap injects exactly the rated MVAR regardless of bus voltage. The simple textbook assumption.
  • const Z (V²B) — the cap is modeled as a constant susceptance B in the Y-bus, so it injects Q ≈ V²·B. A 20 MVAR cap at 0.95 pu injects only ~18 MVAR (= 20 · 0.9025). This is what real shunt capacitors actually do — their VAR output drops when you most need it (low V) and rises when you don't (high V).
• Q reactor (MVAR) — reactive absorption at the bus, also entered as a positive number.
• Reactor model — same options as Cap model. Constant-MVAR reactors absorb their rated value regardless of voltage; constant-Z reactors absorb less at low voltage (which is good for stability) and more at high voltage.
• In service checkboxes — toggle each independently. A bus can have both a cap and a reactor; the net effect is determined by which is in service.

You can mix models freely: a bus can have a constant-MVAR cap and a constant-Z reactor, or vice versa. The choice typically depends on the device technology — switched capacitor banks are well-modeled as constant-MVAR (they're rated at a specific point and switched in/out), while air-core reactors and saturable devices are better modeled as constant-Z. SVCs and STATCOMs in voltage-control mode are something else entirely (active control of Q to hold voltage), beyond what this tool models.

Each transformer row gives:

• Branch / F→T — name and which buses it connects.
• kV ratings — primary and secondary voltage ratings, used to compute the off-nominal turns ratio in conjunction with the bus base kVs.
• r, x — series impedance in per-unit on the system base. b is always 0 for transformers in this tool (no magnetizing branch).
• Tap ratio (a) — off-nominal tap on the from side. The π-equivalent uses V_from/a as the network-side voltage. Increasing a lowers V_to; decreasing a boosts V_to.
• Rating (MVA) — nameplate MVA, used to compute % loading shown in the loading display mode. The fundamental thermal limit is actually a current rating (the windings can only dissipate so much I²R heat); MVA is just the nameplate convention because at 1.0 pu voltage, MVA and current are equivalent. At lower voltage, the same current corresponds to less MVA — so a transformer carrying its nameplate MVA at, say, 0.92 pu is actually carrying about 109% of its rated current and is thermally overloaded even though % loading reads 100%. Switch to the Current (A) display mode to see the actual currents and check this directly against the equipment's nameplate amp rating.
• In service — uncheck to take the transformer out of service. The branch is removed from the Y-bus entirely.
• → PST button — promotes this transformer to a phase-shifting transformer. A default 5° phase shift is applied; the row moves to the PST table where you can adjust it further.

A phase-shifting transformer (PST) is structurally identical to a regular transformer except that it inserts a controllable angle φ between its from and to sides. The complex turns ratio τ = a · e^jφ makes the Y-bus asymmetric — Y_ft ≠ Y_tf when φ ≠ 0. This asymmetry is the entire mechanism: it forces real-power flow from one side to the other regardless of the natural impedance distribution.

The PST row has the same columns as a regular transformer with one addition:

• Phase shift (°) — the angle φ in degrees. Positive means the from-side leads the to-side. Typical values are ±5° to ±20° depending on the device design. You can also drag the corresponding slider in the Controls panel for fine-grained adjustment.
• Demote button — clears the PST status (zeros the phase shift, removes the PST classification) and moves the row back to the regular Transformers table.

Three ways to add a PST:

1. Click the Add Phase Shifting Transformer button in the diagram toolbar, then click two buses to connect them. A default 5° phase shift is applied.
2. Click → PST on any existing transformer row to promote it.
3. Edit a JSON model file with a non-zero phaseShift on a transformer branch.

Real-world example: PSTs are commonly used at the New York/Pennsylvania interface to control loop flows from PJM into NYISO. The interconnection has parallel paths through both control areas; the PSTs let operators dispatch real power along the contractual path rather than letting it follow the natural impedance distribution.

To see the effect in this demo: T4 (the PST tie between B5 and B6) is in parallel with the L57+L67 path through B7. Sweep its phase shift from −15° to +15° and watch the flow on T4 reverse direction while L57 and L67 mirror the change. Total system real-power balance is preserved at every angle — the PST just redirects flow between the two parallel paths.

Each line row gives:

• r (pu) — series resistance in per-unit on the system base.
• x (pu) — series reactance in per-unit. Together with r, this defines the line's series impedance.
• b (pu) — total shunt charging susceptance in per-unit. Half is placed at each end in the π-equivalent. Set to 0 for short lines where charging is negligible.
• Base kV — the line's nominal voltage. Editing this triggers a per-unit rebasing of r, x, b on this and all other lines in the same connected voltage zone, preserving physical ohms.
• Rating (MVA) — thermal rating used for % loading display. As with transformers, the underlying limit is really a current rating (conductor ampacity — how many amps the wire can carry without exceeding its annealing or sag temperature). MVA = √3 · V_LL · I_A at 1.0 pu, and the conversion only stays clean at 1.0 pu. At lower voltage, the same conductor current draws less MVA, so a 100 MVA line carrying its nameplate MVA at 0.95 pu is actually carrying about 105% of its rated current. Use the Current (A) display mode when low-voltage operation matters for ampacity verification.
• In service — same as transformer in-service.

The "P loss" and "Q loss" columns can be negative for lightly-loaded long lines. That's not a bug — it means the line's shunt charging (V²·b/2 at each end) exceeds the I²X consumption in the series reactance. The line is acting as a net VAR source rather than a sink.

This table edits raw bus parameters that don't fit elsewhere:

• Base kV — the bus's voltage zone. Editing rebases all connected branches to preserve physical ohms.
• Pl, Ql — the bus's base load (before scaling and shunt adjustments). Set both to 0 to remove the bus from the Loads table.
• Q cap, Q reactor — the rated MVAR values for shunt compensation devices at this bus.
• B shunt (pu) — a generic shunt susceptance value, separate from the cap/reactor model. Use this if you want to model arbitrary shunt admittance (e.g., line-end equivalents from external systems).

Transformer tap ratios

One slider per transformer. Range typically 0.85–1.15. Changing a tap immediately re-solves and you can watch flows redistribute.

Generator setpoints

Sliders are dynamically generated based on bus type:

• Slack: V setpoint slider only (1.00–1.10 pu typically).
• PV: V setpoint + P dispatch sliders.
• PQ generator: P + Q dispatch sliders.

Slider ranges respect Pmin/Pmax/Qmin/Qmax from the Generators table when they're set. If you type a value into a table cell that's outside the slider's current range, the slider auto-widens.

Load scaling

A single multiplier (0.2× to 1.5×) applied uniformly to every load in the system. Useful for "watch what happens at heavy load" scenarios without editing each bus individually.

Modeling options

• Enforce PV generator Q limits — when checked, the solver runs an outer loop. After each inner solve, any PV bus whose computed Q exceeds Qmin/Qmax converts to PQ with Q clamped at the violated limit; its V then floats to whatever the system can support. The user's PV designation and V setpoint are preserved across the conversion.

Per-device modeling choices (load model, cap/reactor model) live in their respective tables, not here.

Solve / Reset

Solve forces an immediate re-solve (normally the solver runs automatically on any change). Reset returns transformer taps to 1.0 and load scale to 1.0; generator setpoints and topology are not touched.

Export animation

Capture the live flow animation as either an Animated GIF or an MP4/WebM video. Format, aspect ratio, and resolution are all configurable. Output is watermarked with the NERX Power Consultants logo at the bottom-right and the model name at the top.

The ⛶ Fullscreen button (top right of the View toolbar above the diagram) expands the diagram to fill the entire screen. Useful for live presentations and LinkedIn demos where the constrained pane is too small to read at audience distance.

While fullscreen, the diagram and the Branch labels radio buttons are visible but the Generator/Load tables and Controls sliders are not — those stay in the underlying page. So the workflow is: set up your scenario first (adjust load scale, generator dispatch, transformer taps in the underlying view), then click the Fullscreen button. Once fullscreen, you can still toggle between display modes (P/Q flow, current, loading, voltage heatmap, angle heatmap) and zoom/pan the diagram.

Press Esc to exit fullscreen, or click the button again. Re-entering fullscreen always re-fits the diagram to the new pixel dimensions, so the content fills the screen edge-to-edge regardless of monitor aspect ratio.

• P, Q flow (MW, MVAR) — the default. Branch labels show real and reactive flow magnitudes. Branch coloring is neutral.
• Current (A) — branch labels show the current magnitude in amperes at each end of the branch, computed from |I| = |S| / V converted to amps using each bus's base kV (I_A = I_pu × S_base / [√3 × V_base_LL]). For a transmission line both ends carry essentially the same current; for a transformer the two ends differ by the turns ratio (HV side has lower amps, LV side has higher amps). This is the view a relay engineer or protection settings person works in — gear is rated and CT-tapped in amps, not MVA. Equipment thermal limits are fundamentally current limits; MVA is just a convenient nameplate at 1.0 pu voltage.
• Loading (% of rating) — branches recolor by % of their MVA rating: green below 50%, light green 50–80%, amber 80–100%, red above 100%. Useful for spotting overloads at a glance. Caveat: equipment ratings are fundamentally current-based (thermal limits on conductor or winding heat dissipation), and the MVA rating is only equivalent to the current rating at 1.0 pu voltage. At depressed voltage the same nameplate MVA corresponds to a higher current — a transformer reading 100% MVA loading at 0.93 pu is actually carrying about 108% of its rated amps. For a true thermal check at low voltage, switch to the Current (A) display mode and compare against the equipment's nameplate amp rating.
• Bus voltage heatmap — bus rectangles recolor by |V| in pu. Diverging blue→white→red scale centered at 1.0 pu. The two halves are scaled independently so the worst-low bus always saturates blue and the worst-high bus always saturates red. Legend at bottom-left of the diagram.
• Bus angle heatmap — bus rectangles recolor by θ in degrees. Diverging purple→white→orange centered at 0° (slack reference). Same independent-half scaling.

Power flow non-convergence is one of the harder problems in steady-state analysis because it can mean fundamentally different things. When a solve fails, this tool runs an automatic diagnostic that tries to figure out which kind of failure has occurred and tells you what to do about it. The findings appear in a panel below the status bar.

The diagnostic checks for several distinct failure modes:

• Topology error (islanded buses). A bus has no in-service path to the slack bus, so the Y-bus is structurally singular. The diagnostic walks the connectivity graph from the slack and lists every bus it can't reach. Fix: restore an out-of-service branch, or remove the disconnected components.
• Data inconsistency. Things like Qmin > Qmax (no feasible point), Pmin > Pmax, or branches with effectively zero impedance (which would give an infinite admittance). Fix: correct the inconsistent value.
• Bad initial guess. A solution exists but the solver couldn't find it from where it started. Most commonly this happens after a big topology change when the hot-start values mislead the solver. The diagnostic automatically retries from a flat start (V = 1.0) and a warm start (V = 0.95). If either succeeds, the system is left in the recovered state with a green "✓ Recovered" message.
• Voltage collapse. The system is loaded past its voltage-stability limit (the "nose" of the P-V curve). No real solution exists at the requested loading. The diagnostic detects this by binary-searching the load scale: if the system converges at, say, 60% of the requested load but not at 100%, you're past the nose at 100%. The panel reports the approximate nose location. Fix options: reduce loads, add reactive support (caps/SVCs at the weak buses), strengthen transmission paths, or raise generator V setpoints.
• Q-limit chattering. With Q-limits enforcement enabled, the outer loop can sometimes oscillate between two configurations of which buses are clamped — this means no fixed-point assignment exists with the current limits. The diagnostic detects when the same conversion pattern repeats and stops, rather than spinning forever.

The auto-recovery is silent on success — if the only thing wrong was a stale hot-start, you'll see the "✓ Recovered" note and the model continues. Only persistent failures show the full diagnostic panel.

Voltage collapse is the most pedagogically interesting of these. In real grids it's the regime where utilities deploy SVCs, STATCOMs, switched capacitor banks, or under-voltage load shedding. The "P-V curve" or "nose curve" is the canonical visualization — the diagnostic's nose-scale estimate tells you roughly where the tip is. To explore this, try cranking individual load values up in the Loads table (the load-scale slider only goes to 1.5×, but the table cells accept any value).

When voltage collapse is detected, a "Trace P-V curve" button appears in the diagnostic panel. Clicking it runs an incremental-loading procedure (a simplified continuation power flow): starting from a low-load operating point, the load is gradually scaled up and the system re-solved at each step, hot-starting from the previous solution. Step size shrinks automatically as voltage drop accelerates near the nose. The result is plotted as |V| vs. λ for each load bus — that's the actual upper branch of the P-V curve for your system. The right edge marks the nose location with reasonable precision.

Other techniques baked into the solver to improve robustness:

• Damped Newton step. A full Newton step can overshoot near folds or with bad starting points. Each step is capped to keep |Δθ| ≤ 30° and V ∈ [0.4, 1.6]. Near a real solution the cap doesn't bind, so converged solves take the same number of iterations as undamped Newton; far from a solution, damping prevents the iterates from blowing up.
• DC power flow seed. On cold starts, the solver first solves the linearized lossless DC equations to get a better angle starting point than θ = 0. For systems with heavy transmission flow (where real angles can spread 20–30°), this saves 1–2 iterations and rescues some cases where flat θ = 0 would lead Newton astray.

This tool implements a standard balanced positive-sequence AC power flow with the following modeling choices:

• Solver: Newton-Raphson in polar form, with full Jacobian and partial-pivot Gaussian elimination. Damped step control caps |Δθ| at 30° and keeps V within [0.5, 1.5] per iteration to prevent overshoot near folds. Hot-start from previous solution for incremental edits; DC power flow seed on cold starts (solves the linearized lossless equations B'·θ = P first to get a much better θ starting estimate than θ = 0, especially for systems with significant transmission flow).
• System base: 100 MVA. All per-unit values are on this base.
• Convergence: max mismatch tolerance 1×10⁻⁶ pu, with a 50-iteration cap.
• Lines: nominal-π model with series r + jx and total shunt jb (half at each end). No long-line correction. No zero-sequence or harmonic modeling.
• Transformers: off-nominal-tap π-equivalent, from-side tap convention. No magnetizing branch. Phase-shifting transformers supported (complex-tap model τ = a·e^(jφ)). Charging b is supported but typically zero for transformers.
• Loads: per-bus selectable constant power (default) or constant impedance (P, Q ∝ V²). No constant-current, no full ZIP, no motor models.
• Shunts: per-device selectable constant-MVAR (default) or constant-susceptance. Cap and reactor at the same bus can use different models.
• Generator Q limits: optional, enforced via PV→PQ conversion outer loop.
• Generator impedance: optional Thevenin series Zgen = r + jx per generator. Implemented as a phantom internal bus during the solve, transparent to the rest of the UI.
• What's not modeled: dynamic stability, fault analysis, harmonics, three-phase asymmetry, frequency dependence, voltage-dependent loads, contingency screening, optimal power flow.

For an 8-bus demo, this is more than adequate. For larger or more rigorous study work, use a commercial tool — but the math here is the same math those tools run.