Formula sheet

Power Flow and Grid Stability Formula Sheet

Formula sheet for power flow and grid stability covering per-unit bases, three-phase power, line loading, voltage drop, reactive power, short-circuit screening, inertia, RoCoF, frequency response, and stability margins.

Branch: Energy Engineering
Content: Formula sheet
Updated: Jun 20, 2026
Revision: v1.0.0 · reviewed

This formula sheet collects first-pass equations used in power-flow screening and grid-stability review. Use it for concept studies, interconnection checks, operating envelopes, commissioning acceptance, and engineering exercises. Detailed studies still require load-flow software, short-circuit models, protection coordination, dynamic simulation, electromagnetic transient models, validated controls, and current grid-code requirements.

State the boundary before using any formula: generator terminals, inverter AC terminals, transformer high-voltage side, collector bus, distribution feeder, transmission bus, or point of interconnection. A number is not useful unless the voltage base, power base, operating case, and measurement boundary are known.

Per-Unit Bases

Per-unit value:

\displaystyle x_{pu}=\frac{x_{actual}}{x_{base}}

Three-phase apparent-power base:

S_{base,3\phi}=\sqrt{3}V_{LL,base}I_{base}

Line current base:

\displaystyle I_{base}=\frac{S_{base,3\phi}}{\sqrt{3}V_{LL,base}}

Impedance base:

\displaystyle Z_{base}=\frac{V_{LL,base}^2}{S_{base,3\phi}}

Admittance base:

\displaystyle Y_{base}=\frac{1}{Z_{base}}

Per-unit conversion helps compare equipment at different voltage levels, but the base values must be stated. A transformer, cable, inverter, and generator can all use different nameplate bases before conversion.

Balanced Three-Phase Power

Apparent power:

S=\sqrt{3}V_{LL}I_L

Active power:

P=\sqrt{3}V_{LL}I_LPF

Reactive power:

Q=\sqrt{S^2-P^2}

Power factor:

\displaystyle PF=\frac{P}{S}

These equations assume balanced sinusoidal operation. Harmonics, unbalance, faults, saturation, converter current limits, and transients require additional models.

Line and Transformer Loading

Loading fraction:

\displaystyle L=\frac{S_{case}}{S_{rating}}

Percent loading:

\displaystyle L_{\%}=100\frac{S_{case}}{S_{rating}}

Current margin:

M_I=I_{rating}-I_{case}

Apparent-power margin:

M_S=S_{rating}-S_{case}

For an N-1 screen with one largest unit unavailable:

S_{firm,N-1}=\sum_i S_i-\max(S_i)

This checks capacity only. It does not prove protection, thermal time constants, cooling, transformer overload rules, maintenance access, or physical separation.

Simplified Voltage Drop

Approximate three-phase voltage drop for a line or feeder:

\displaystyle \Delta V\approx \frac{R P+X Q}{V}

Per-unit form:

\Delta V_{pu}\approx R_{pu}P_{pu}+X_{pu}Q_{pu}

where $R$ and $X$ are series resistance and reactance at the selected base. This screening relation shows why reactive power can strongly affect voltage, especially on inductive feeders.

Voltage-rise checks for distributed generation should use the correct sign convention. Exporting active power on a resistive feeder can raise local voltage; absorbing or injecting reactive power can either reduce or increase voltage depending on network impedance and control mode.

Reactive Power Capability

Apparent-power limit:

S_{max}^2=P^2+Q^2

Reactive capability at active power $P$ :

|Q|_{max}=\sqrt{S_{max}^2-P^2}

Active-power capability at reactive demand $Q$ :

P_{max}=\sqrt{S_{max}^2-Q^2}

Converter and generator capability curves may be more restrictive than this circle because of current limits, voltage limits, thermal limits, excitation limits, DC-link limits, grid-code modes, or control settings.

Approximate Active-Power Transfer

For a simple lossless line between buses with voltage magnitudes $V_1$ and $V_2$ and reactance $X$ :

\displaystyle P\approx \frac{V_1V_2}{X}\sin\delta

For small angle difference:

\displaystyle P\approx \frac{V_1V_2}{X}\delta

where $\delta$ is in radians. This expression explains why power angle, voltage magnitude, and transfer reactance matter for synchronism and transmission loading.

It is a teaching approximation. Real grids include resistance, voltage control, multiple paths, limits, protection, generator dynamics, load response, and converter controls.

Short-Circuit Screening

Three-phase fault current from Thevenin impedance:

\displaystyle I_{sc}=\frac{V_{LL}}{\sqrt{3}Z_{th}}

Short-circuit MVA:

S_{sc}=\sqrt{3}V_{LL}I_{sc}

Short-circuit ratio for an inverter or plant:

\displaystyle SCR=\frac{S_{sc}}{S_{plant}}

Lower SCR indicates a weaker grid for the plant size. Weak grids can increase voltage sensitivity, control interaction, ride-through difficulty, and harmonic or stability risk. SCR is a screen, not a complete stability proof.

Frequency Deviation from Power Imbalance

A power imbalance:

\Delta P=P_{generation}-P_{load}

If generation is lower than load, $\Delta P$ is negative and frequency tends to fall.

A simplified system swing relation is:

\displaystyle \frac{df}{dt}\approx \frac{f_0}{2H_{sys}S_{base}}\Delta P

where:

$f_0$ is nominal frequency;
$H_{sys}$ is equivalent inertia constant;
$S_{base}$ is system power base;
$\Delta P$ is active-power imbalance on the same base.

This relation is a first screen for rate of change of frequency. Real frequency behavior also depends on governors, inverter controls, load damping, protection, reserves, and measurement filtering.

Inertia Constant

Stored kinetic energy in rotating mass:

E_k=H S_{rated}

where $H$ is inertia constant in seconds and $S_{rated}$ is machine apparent-power rating.

For multiple synchronous machines on a common base:

\displaystyle H_{eq}=\frac{\sum_i H_i S_i}{\sum_i S_i}

Inverter-based resources do not automatically provide synchronous inertia. They may provide fast frequency response, synthetic inertia-like control, or grid-forming behavior, but those services must be specified and validated.

Primary Frequency Response

Governor or droop relation:

\displaystyle \Delta P=-\frac{1}{R}\Delta f

where $R$ is droop expressed in power per frequency unit, depending on convention.

Percent droop is often interpreted as the frequency change required to move from no-load to full-load power. A smaller droop value means a stronger power response to frequency deviation.

For several responding resources, approximate aggregate response is:

\Delta P_{response}=\sum_i \Delta P_i

Response must be checked against ramp rate, deadband, measurement delay, headroom, energy reserve, converter current limit, and control priority.

Voltage Stability Screen

A simple reactive margin is:

M_Q=Q_{available}-Q_{required}

A positive margin does not prove voltage stability, but a small or negative margin is a warning. Voltage stability depends on load behavior, line impedance, transformer taps, generator excitation, converter controls, capacitor banks, reactors, and contingency state.

A practical voltage-support review should include:

high-load and low-load voltage cases;
maximum generation and minimum demand;
transformer tap positions;
reactive capability of generators and inverters;
capacitor or reactor switching steps;
N-1 network outages;
control interactions and time delays.

Damping and Oscillation Screen

For a lightly damped mode with damping ratio $\zeta$ and natural frequency $\omega_n$ :

\omega_d=\omega_n\sqrt{1-\zeta^2}

Settling time approximation:

\displaystyle t_s\approx \frac{4}{\zeta\omega_n}

In power systems, poorly damped oscillations can appear after disturbances or control interactions. Modal analysis, dynamic simulation, and field measurements are required for serious assessment. The formula is a screening reminder that damping and time scale both matter.

Harmonic Distortion

Total harmonic distortion for voltage:

\displaystyle THD_V=\frac{\sqrt{\sum_{h=2}^{n}V_h^2}}{V_1}

Total harmonic distortion for current:

\displaystyle THD_I=\frac{\sqrt{\sum_{h=2}^{n}I_h^2}}{I_1}

Power electronic converters, filters, transformers, cables, and weak grids can interact. Harmonic compliance should be tested at the point and operating conditions specified by the interconnection requirement.

Validation Evidence

Useful validation checks include:

Claim	Evidence
Load-flow model is credible	Metered voltage, current, active power, and reactive power compared with model.
Reactive capability is sufficient	Tested $P$ - $Q$ operation at representative voltage and temperature.
Short-circuit assumptions are valid	Utility fault-level data, equipment model, and protection-study records.
Frequency response is available	Event record or staged test showing response magnitude and timing.
Voltage control is stable	Step response, tap operation, inverter control logs, and no sustained oscillation.
Harmonic performance is acceptable	Power-quality measurement at the required boundary.

The equations in this sheet are screens. They help identify what must be studied in more detail and what evidence must be collected before a grid claim is accepted.

Common Mistakes

A common mistake is using MW alone for grid integration. Current, apparent power, reactive power, voltage, short-circuit level, inertia, controls, and protection can dominate the real limit.

Another mistake is using a strong-grid assumption after adding inverter-based generation to a weak network. Converter current limits, phase-locked loops, grid-forming modes, filters, and protection settings can change system behavior.

A deeper mistake is treating a passed load-flow case as proof of stability. Load-flow analysis is steady state. Stability requires dynamic evidence over the disturbances, controls, and protection actions that matter.

REF

Disciplines