Formula sheet

Ship Power Systems and Propulsion Plant Formula Sheet

Ship power formulas for generator reserve, AC loading, diesel-electric losses, shaft power, fuel, cooling, batteries, load shedding, reliability, and release checks.

Branch: Naval and Marine Engineering
Content: Formula sheet
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects first-pass calculations for ship power systems and propulsion-plant integration. Use it to screen generator loading, electrical quantities, diesel-electric conversion losses, shaft power, fuel demand, cooling duty, battery support, load shedding, reliability, and release evidence.

It complements the vessel-performance formula sheet. That page covers hydrostatics, resistance, propeller performance, cavitation, and vessel response. This page focuses on the plant boundary: generators, propulsion drives, motors, switchboards, batteries, cooling systems, auxiliaries, emergency power, controls, and operating modes.

The formulas are educational screening tools. Real vessels require class and flag requirements, approved electrical studies, short-circuit and protection coordination, harmonic studies, machinery ratings, fire-zone separation, redundancy notation, sea-trial data, crew procedures, and manufacturer limits.

How to Use This Formula Sheet

Use this sheet to connect ship electrical power, propulsion demand, batteries, emergency loads, cooling, fuel, redundancy, and release evidence for a defined operating mode. Start by defining the vessel mode, power boundary, online generators, switchboard arrangement, propulsion load, auxiliary load, emergency load, battery or UPS state, cooling state, redundancy requirement, and limiting failure case. Then decide whether the calculation supports mode release, sea trial review, blackout recovery, retrofit design, or troubleshooting.

Work through the checks in this order:

Establish shaft, motor, drive input, generator output, bus, battery, fuel, and cooling boundaries before comparing powers.
Check active power, reactive power, apparent power, current, harmonic limits, online reserve, N-1 capacity, load shedding, and frequency-dip response for the operating mode.
Check propulsion conversion losses, torque, propeller load law, fuel endurance, cooling duty, battery support, UPS duration, load sharing, and reliability under normal and degraded states.
Compare calculated margins with sea-trial data, switchboard measurements, power-management logs, battery telemetry, fuel-flow records, cooling-flow data, blackout-recovery tests, and crew procedure evidence.
State the release action: accept mode, restrict load, revise load-shed table, adjust PMS logic, add cooling reserve, derate operation, or require further validation.

Do not claim survivable power from installed nameplate power. The vessel only has the power that is online, available, derated correctly, synchronized, cooled, protected, and allowed by the operating mode after the limiting failure.

Basis and Validity Limits

The formulas below are first-pass screens. They assume that the operating mode, power boundary, redundancy basis, derating, load list, failure case, and acceptance criterion are defined.

Electrical loading formulas are valid only when voltage, frequency, power factor, harmonic distortion, cable rating, switchboard rating, protection settings, generator capability curve, cooling condition, and installation derating are considered.

Propulsion and fuel formulas are valid only when shaft power, bus power, drive losses, motor efficiency, gearbox efficiency, propeller operating point, auxiliary load, sea state, maneuvering mode, and hotel load are separated. A calm-water shaft estimate is not a full plant load case.

Battery, UPS, load-shedding, and blackout-recovery formulas are valid only when state of charge, state of health, thermal limit, converter rating, fire safety, repeated-event capability, sequencing, timing, interlocks, emergency loads, and crew response are validated.

Power boundaries

Always state the boundary before calculating.

Boundary	Meaning	Typical use
$P_{shaft}$	mechanical power at propeller or shaft line	shafting, propulsion demand, sea trial
$P_{motor}$	mechanical motor output	motor sizing, torque checks
$P_{drive,in}$	electrical input to propulsion drive	switchboard loading
$P_g$	generator electrical output	online capacity and reserve
$P_{fuel}$	chemical fuel power	endurance and efficiency
$\dot{Q}_{cool}$	heat rejected to cooling systems	thermal release

Do not mix shaft power, bus power, generator output, and fuel power. Conversion losses and auxiliary loads decide whether a plant with enough installed power can actually support the operating mode.

Active, reactive, and apparent power

Single-load apparent power:

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

Reactive power:

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

Three-phase active power:

P=\sqrt{3}V_{LL}I PF

Three-phase apparent power:

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

Line current:

\displaystyle I=\frac{P}{\sqrt{3}V_{LL}PF}

where $V_{LL}$ is line-to-line voltage. Use consistent units: watts, volts, amperes, volt-amperes, and dimensionless power factor.

Low power factor can overload generator and transformer apparent-power capacity even when active kilowatts appear acceptable. Harmonic distortion can further heat equipment and disturb protection or control.

Online capacity and reserve

Online active-power capacity:

P_{online}=\sum_i P_{g,i} d_i

where $d_i$ is a derating factor for ambient temperature, cooling limitation, fuel mode, altitude, maintenance restriction, or class-approved operating limit.

Total active load:

P_L=\sum_j P_j

Spinning reserve:

P_R=P_{online}-P_L

Reserve fraction:

\displaystyle R_f=\frac{P_R}{P_{online}}

Available reserve is not only installed nameplate power. A generator that is unavailable, derated, not synchronized, cooling-limited, fuel-limited, or blocked by protection logic is not part of the online reserve.

N-1 contingency

Surviving capacity after the largest online source is lost:

P_{N-1}=P_{online}-P_{largest}

Allowed temporary load after contingency:

P_{allowed}=f_{limit}P_{N-1}

Required load reduction:

P_{shed,req}=\max(0,P_L-P_{allowed})

where $f_{limit}$ is the permitted loading fraction during recovery.

The same check should be repeated for the limiting contingency: generator trip, bus-tie trip, propulsion drive trip, cooling-pump loss, fuel-transfer failure, control-network loss, or emergency switchboard transition.

Load priority and shedding

Total planned shedding:

P_{shed,plan}=\sum_k P_{shed,k}

Remaining load after shedding:

P_{remain}=P_L-P_{shed,plan}

Shed margin:

M_{shed}=P_{allowed}-P_{remain}

If $M_{shed}<0$ , the load-shedding table is too small for the claimed operating mode.

Timing matters. A stage that acts after frequency or voltage has already crossed a protection threshold may not prevent blackout. Load-shedding logic should be tied to generator-trip, overload, bus, frequency, voltage, and priority evidence, not only to delayed alarms.

Frequency-dip screen

A simplified early-time frequency decline after active-power imbalance is:

\displaystyle \frac{df}{dt}\approx-\frac{f_0}{2H}\frac{\Delta P}{S_b}

where $f_0$ is nominal frequency, $H$ is inertia constant on the selected base, $\Delta P$ is active-power deficit, and $S_b$ is the machine or system base power.

Approximate frequency drop over a short delay:

\displaystyle \Delta f\approx \left|\frac{df}{dt}\right|\Delta t

This is a screening relation, not a dynamic power-system simulation. Real response depends on governors, automatic voltage regulation, motor load dynamics, drive controls, protection settings, battery support, and load-shedding timing.

Diesel-electric conversion losses

Overall electrical-to-shaft efficiency:

\eta_{e\rightarrow s}=\eta_{drive}\eta_{motor}\eta_{gear}\eta_{shaft}

Bus power needed for shaft demand:

\displaystyle P_{bus}=\frac{P_{shaft}}{\eta_{e\rightarrow s}}

Total conversion loss:

P_{loss}=P_{bus}-P_{shaft}

For multiple propulsion drives:

\displaystyle P_{prop,bus}=\sum_i \frac{P_{shaft,i}}{\eta_i}

Use the correct operating point. Drive, motor, gearbox, and propeller efficiencies are not constants across speed, torque, cooling condition, and control mode.

Shaft power, torque, and speed

Angular speed:

\omega=2\pi n

where $n$ is rotations per second.

Shaft power:

P=\tau\omega

Torque:

\displaystyle \tau=\frac{P}{\omega}

Gear ratio:

\displaystyle i=\frac{\omega_{in}}{\omega_{out}}

Ideal output torque through a reduction gear:

\tau_{out}\approx \eta_g i \tau_{in}

Torque checks should include transient acceleration, crash stop, clutch engagement, ice or debris events where relevant, torsional vibration, bearing limits, shaft stress, and gearbox thermal limits.

Propulsion load law

A common propeller-like load approximation is:

\displaystyle P_2=P_1\left(\frac{n_2}{n_1}\right)^3

and:

\displaystyle \tau_2=\tau_1\left(\frac{n_2}{n_1}\right)^2

These relations are useful for screening a fixed-pitch propeller near similar operating conditions. They can be poor during maneuvering, bollard pull, dynamic positioning, heavy weather, shallow water, controllable-pitch operation, waterjet operation, or strong wake changes.

Fuel power and endurance

Fuel power:

P_{fuel}=\dot{m}_f LHV

Thermal efficiency:

\displaystyle \eta_{th}=\frac{P_{out}}{\dot{m}_f LHV}

Fuel mass flow from brake-specific fuel consumption:

\dot{m}_f=BSFC\cdot P_{out}

Fuel mass for an operating period:

m_f=\dot{m}_f t

Endurance from usable fuel:

\displaystyle t_{end}=\frac{m_{fuel,usable}}{\dot{m}_f}

State whether fuel consumption is based on engine brake power, generator electrical output, shaft power, or total vessel operating load. The answer changes with auxiliary load, generator loading, battery strategy, hotel load, and operating mode.

Cooling duty

Heat rejected by an inefficient conversion stage:

\dot{Q}_{loss}=P_{in}(1-\eta)

Cooling-water heat balance:

\dot{Q}_{cool}=\dot{m}c_p\Delta T

Required coolant mass flow:

\displaystyle \dot{m}=\frac{\dot{Q}_{cool}}{c_p\Delta T}

Volumetric flow:

\displaystyle Q=\frac{\dot{m}}{\rho}

Heat-exchanger conductance screen:

\dot{Q}=UA\Delta T_{lm}

Cooling release should check seawater temperature, fouling, strainer blockage, pump NPSH, corrosion, redundancy, power supply, alarm thresholds, and whether the cooling system supports degraded propulsion modes.

Battery energy and state of charge

Usable battery energy:

E_{usable}=E_{nom}(SOC_{max}-SOC_{min})SOH

Discharge energy for a support event:

\displaystyle E_{event}=\frac{P_{bat}t}{\eta_{dis}}

State-of-charge change:

\displaystyle \Delta SOC=\frac{E_{event}}{E_{nom}}

C-rate:

\displaystyle C_{rate}=\frac{P_{bat}}{E_{nom}}

Round-trip energy recovered after a cycle:

E_{out}=\eta_{rt}E_{in}

Battery support must be checked against power rating, converter rating, state-of-charge window, thermal limits, fire safety, isolation, degradation, repeated-event capability, ventilation, and class rules. Nameplate energy alone is not enough.

UPS and emergency load duration

Emergency energy required:

E_{req}=\sum_j P_{crit,j}t_j

Available UPS energy:

E_{UPS,usable}=E_{UPS,nom}(SOC_{max}-SOC_{min})\eta_{UPS}

Duration at critical load:

\displaystyle t=\frac{E_{UPS,usable}}{P_{crit}}

Emergency loads should be divided by function: steering, navigation, communications, emergency lighting, fire and bilge systems, control power, shutdown systems, alarms, and essential cooling. A single average emergency load can hide a starting-current or mission-duration problem.

Equal percentage load sharing for online generators:

\displaystyle \frac{P_i}{P_{rated,i}}\approx \frac{P_j}{P_{rated,j}}

Load on generator $i$ under proportional sharing:

\displaystyle P_i=P_L\frac{P_{rated,i}}{\sum_k P_{rated,k}}

Generator utilization:

\displaystyle U_i=\frac{P_i}{P_{rated,i}}

Poor load sharing can overload one generator while total online capacity appears adequate. Validate real load sharing during load steps, thruster changes, battery charge transitions, shore-power transfer, and recovery from standby generator synchronization.

Apparent-power and current margin

Generator apparent-power utilization:

\displaystyle U_S=\frac{S_L}{S_{rated}}

Current margin:

M_I=I_{limit}-I_{meas}

Cable or breaker loading fraction:

\displaystyle U_I=\frac{I_{meas}}{I_{rating}}

Use ratings that match installation temperature, bundling, ventilation, enclosure, harmonic content, duty cycle, and protection settings. A clean current value can be misleading if harmonics or unbalanced loads are significant.

Reliability and availability

Availability of a repairable function:

\displaystyle A=\frac{MTBF}{MTBF+MTTR}

Series availability for functions that all must work:

A_{series}=\prod_i A_i

Parallel availability for redundant alternatives:

A_{parallel}=1-\prod_i(1-A_i)

The function must be stated before applying the formula. “Propulsion available for open-water transit” is different from “station keeping near an offshore installation after one fire-zone loss” or “emergency steering during blackout recovery.”

Risk and validation screens

Risk priority number:

RPN=SOD

Validation agreement:

\displaystyle e=\frac{x_{meas}-x_{calc}}{x_{calc}}

Acceptance with uncertainty:

M=x_{limit}-x_{meas}-ku_x

or, for a minimum required value:

M=x_{meas}-x_{min}-ku_x

where $u_x$ is standard uncertainty and $k$ is a selected guard factor.

Use uncertainty guards for sea-trial shaft power, fuel-flow meters, switchboard measurements, battery state of charge, cooling flow, and load-shed timing when a margin is safety-critical.

Worked screening example

A diesel-electric vessel is preparing for low-speed maneuvering. Two generators are online and one is on standby.

Quantity	Value
Online generators	$2$
Generator rating	$2.0\ \text{MW}$ each
Derating factor	$0.95$
Active load before contingency	$3.10\ \text{MW}$
Largest online generator	$1.90\ \text{MW}$ derated
Permitted temporary loading after one trip	$85\%$ of survivor capacity
Propulsion shaft demand included in load	$1.40\ \text{MW}$
Propulsion conversion efficiency	$0.91$
Line voltage	$690\ \text{V}$
Power factor during maneuvering	$0.86$
Battery support command	$0.60\ \text{MW}$ for $10\ \text{min}$
Battery discharge efficiency	$0.94$

Online capacity:

P_{online}=2(2.0)(0.95)=3.80\ \text{MW}

Spinning reserve:

P_R=3.80-3.10=0.70\ \text{MW}

The reserve is positive in normal operation. Now test the single-generator contingency:

P_{N-1}=3.80-1.90=1.90\ \text{MW}

Allowed temporary load:

P_{allowed}=0.85(1.90)=1.615\ \text{MW}

Required shedding or support:

P_{shed,req}=3.10-1.615=1.485\ \text{MW}

If the load-shedding table removes $0.95\ \text{MW}$ and the battery provides $0.60\ \text{MW}$ quickly enough, total relief is:

P_{relief}=0.95+0.60=1.55\ \text{MW}

Shed margin:

M_{shed}=1.55-1.485=0.065\ \text{MW}

The power margin is only $65\ \text{kW}$ , so timing, measurement uncertainty, battery state of charge, and generator transient response are critical.

Bus power corresponding to the shaft demand:

\displaystyle P_{bus}=\frac{1.40}{0.91}=1.54\ \text{MW}

Three-phase line current for the total pre-contingency load:

\displaystyle I=\frac{3.10\times10^6}{\sqrt{3}(690)(0.86)}=3010\ \text{A}

Battery energy required:

\displaystyle E_{event}=\frac{0.60(10/60)}{0.94}=0.106\ \text{MWh}=106\ \text{kWh}

If the battery operating window and state of health cannot guarantee at least $106\ \text{kWh}$ at the required power and temperature, the claimed contingency response is not released.

Engineering comment

The normal two-generator reserve looks acceptable, but the N-1 calculation is the controlling check. The vessel can claim the maneuvering mode only if load shedding and battery support act fast enough, the remaining generator survives the transient, critical loads remain powered, and the validation record proves the sequence under realistic thruster and auxiliary demand.

Validation Evidence Package

Before releasing a ship power or propulsion-plant operating mode, confirm that:

the power boundary is explicit for every calculation;
online generator capacity uses actual derating and availability;
N-1 or required redundancy is checked against the limiting failure;
active, reactive, apparent power, current, and harmonic limits are acceptable;
load shedding is large enough and fast enough for the contingency;
battery or UPS support is inside power, energy, state-of-charge, thermal, and repeatability limits;
propulsion conversion losses are included in bus load and cooling duty;
fuel endurance uses the correct load and efficiency boundary;
cooling systems support the operating and degraded modes;
protection, interlocks, alarms, and manual recovery steps have validation evidence.
sea-trial, dock trial, blackout-recovery, load-step, PMS, battery, switchboard, fuel-flow, and cooling-flow records support the claimed mode.
the release statement identifies the vessel mode, weather or mission condition, failure case, redundancy basis, operator action, and retest trigger.

Common Formula Mistakes

Common mistakes include treating installed generator power as online survivable power, ignoring apparent-power limits, using shaft power as if it were bus power, relying on battery nameplate energy without state-of-charge and power checks, and validating a load-shed table by total megawatts without checking timing.

Another frequent mistake is separating propulsion, electrical distribution, cooling, and controls. A vessel may fail a maneuvering or station-keeping mode because a cooling pump trips, a switchboard section is overloaded, an interlock blocks restart, or a nonessential load is not shed. The useful calculation is the one tied to the operating mode, failure case, measured evidence, and release authority.

Also avoid using one normal operating point for all modes. Harbor maneuvering, DP, open-water transit, emergency steering, blackout recovery, shore-power transfer, battery support, and fire-zone loss can produce different controlling loads and different release evidence.

REF

Disciplines

Ship Power Systems and Propulsion Plant Formula Sheet

How to Use This Formula Sheet

Basis and Validity Limits

Power boundaries

Active, reactive, and apparent power

Online capacity and reserve

N-1 contingency

Load priority and shedding

Frequency-dip screen

Diesel-electric conversion losses

Shaft power, torque, and speed

Propulsion load law

Fuel power and endurance

Cooling duty

Battery energy and state of charge

UPS and emergency load duration

Apparent-power and current margin

Reliability and availability

Risk and validation screens

Worked screening example

Engineering comment

Validation Evidence Package

Common Formula Mistakes

See also

How to Use This Formula Sheet

Basis and Validity Limits

Power boundaries

Active, reactive, and apparent power

Online capacity and reserve

N-1 contingency

Load priority and shedding

Frequency-dip screen

Diesel-electric conversion losses

Shaft power, torque, and speed

Propulsion load law

Fuel power and endurance

Cooling duty

Battery energy and state of charge

UPS and emergency load duration

Generator load sharing

Apparent-power and current margin

Reliability and availability

Risk and validation screens

Worked screening example

Engineering comment

Validation Evidence Package

Common Formula Mistakes