Supercapacitor–Battery Hybrid Sizing for Cranes and Rail

The load is a repeating DC-bus profile: positive motoring and negative regeneration, with constant auxiliary demand and external supply. Hoist pulses have raised-cosine ramps and a plateau; rail pulses are triangular. All entered powers are at the DC bus, so drivetrain efficiency must already be included.

The periodic first-order split solves $\tau\,dP_\mathrm{slow}/dt+P_\mathrm{slow}=P_\mathrm{load}-P_\mathrm{external}$, with $P_\mathrm{slow}(T)=P_\mathrm{slow}(0)$. Closed-form pulse responses solve this same differential equation without missing short pulses. The cutoff is $f_c=1/(2\pi\tau)$. A constant recharge bias $b$ gives $P_\mathrm{SC}=P_\mathrm{net}-P_\mathrm{slow}-b$ and $P_\mathrm{battery}=P_\mathrm{slow}+b$.

For $N_s$ modules in series and $N_p$ strings in parallel, $C=C_\mathrm{module}k_C N_p/N_s$ and $R=R_\mathrm{module}k_R N_s/N_p$. The usable electrostatic window is $\tfrac12C(V_\mathrm{max}^2-V_\mathrm{min}^2)$, reduced by the entered reserve. Directional converter efficiency maps bus power to terminal power $P_t$. The physical current root is $I=2P_t/(V+\sqrt{V^2-4RP_t})$; $dE/dt=-VI$, $V=\sqrt{2E/C}$, and ESR heat is $I^2R$.

A relaxed integral iteration solves voltage and recharge bias together, enforcing zero net capacitor energy per duty. The trajectory is placed at the highest permitted energy to reduce losses while keeping reserve at both limits. At each time, the internal voltage ceiling is also constrained by $V_\mathrm{rated}+RP_t/V_\mathrm{rated}$, so charging ESR rise cannot push the module terminal voltage above its working rating. Doubling and integer bisection select the smallest feasible parallel count, checking the energy window, current discriminant, module peak/RMS currents, and working terminal voltage.

Battery terminal power includes its converter efficiency. Its cumulative energy excursion over one duty plus $(N-1)|\Delta E|$ gives the repeated-duty window. Nominal battery energy is the largest of this window divided by usable SOC, discharge power divided by allowed discharge C-rate, and charge power divided by allowed charge C-rate. Converter ratings cover both ports' peak power with the chosen margin; current duties and storage-port voltage extremes are reported separately.

Braking recovery integrates net negative load after auxiliaries. When external supply or internal cross-flow also charges storage, braking is attributed in proportion to its share of total bus charging power. The reported recovered energy is the resulting increase in stored energy after charging losses.

Battery equivalent full cycles are total terminal-energy throughput divided by twice nominal energy, including the energy needed to reset the battery between duty blocks. Dividing the user-rated full-cycle budget by annual equivalent cycles gives a throughput-life screen, capped by calendar life. Supercapacitor life divides its user-rated duty budget by annual duties, with its own calendar cap.

• The entered duty repeats identically. The capacitor starts at the reported periodic initial voltage, and the battery is reset to the same SOC after the entered number of duties.
• The DC bus is stiff; storage converters are bidirectional and may buck or boost over the reported port-voltage range. A single shared converter is not assumed.
• Identical modules share voltage and parallel current equally. Aged capacitance and ESR are constant over a duty; leakage, balancing consumption and interconnect resistance are excluded.
• Battery voltage is fixed at its nominal value. Converter efficiency is constant in each direction; battery internal losses, voltage sag and SOC-dependent charge acceptance are not resolved.
• Pulse durations and ramp fractions are user-defined examples, not reproductions of measured crane or railway routes. The crane paper informs the power scale; its state-machine controller is not implemented. This tool implements the requested first-order frequency split.
• Battery-only relief uses the same battery energy, voltage, external supply and efficiency as the hybrid design. Peak and RMS relief can be negative for unfavorable settings.

Use measured DC-bus power profiles and component limits for a design decision. The two pulse shapes do not model route gradients, crane mechanics, multiple simultaneous moves, train interactions, catenary receptivity or a supervisory dispatch optimizer. Constant external supply may cause battery charging during idle; a positive net battery energy per duty requires later recharge, while a negative value requires an energy sink or reduced supply.

Sizing uses 600 cosine-spaced integration intervals per nonzero profile phase (4801 samples at the defaults). Reported extrema, energy integrals and current duties use that solve. Displayed curves use a separate 1921-point mesh; capacitor energy is sampled from the finer solve with shape-preserving PCHIP interpolation. The filter response is analytic; ESR and stored energy use cumulative trapezoidal integration with a relative cycle-closure tolerance of 10⁻⁶. A 9601-point refinement exceeds the approximately three-second authoring budget; the 4801-point solve retains the same physics and changes the key default sizes and recovered energy by less than 0.01% relative to that refinement. Integer module counts may change near a feasibility boundary. The search stops at 4096 parallel strings; nonconvergent or electrically infeasible candidates are rejected. Numerical reserve is not a guarantee against an unmeasured surge.

The module pulse-current limit must be valid for the actual pulse duration in both directions. RMS current is averaged over the complete entered duty. Check cooling, transient temperature, cell balancing, cabling, series insulation and aged current ratings separately. The DC/DC power multiplier does not replace switching-ripple, magnetic, semiconductor, control-loop, short-circuit or thermal design; apply it to the reported current and voltage duties when selecting hardware.

Recovered energy is a charging-side energy quantity, not a round-trip energy saving, fuel-saving prediction, stopping-distance result or braking-performance rating. Storage may be unavailable or full: brake choppers, friction brakes and fault operation need separate design.

Life outputs are budget screens, not ageing predictions or warranties. Battery full-cycle ratings must apply to the chosen chemistry, SOC range, temperature and C-rates. Reset charging is included in throughput, but its duration and converter power are not sized. A counted supercapacitor duty is meaningful only when the supplier's cycle test covers the entered waveform; voltage, temperature and dwell time can govern earlier. Installation work belongs to a qualified person.