parametric_model.md

Reverse-Engineered Parametric Performance Model

This document describes the closed-form parametric model reverse-engineered from the SWOPP3 performance model binary (swopp3_performance_model).

Reference Vessel

The SWOPP3 performance model is calibrated for a specific vessel:

Property	Value
Type	Single-skeg general cargo ship
Length overall ($L$)	88 m
Estimated beam ($B$)	~15 m
Estimated wetted surface ($S$)	~2200 m²
Propulsion	CPP, electric
Wingsails	4 × 138 m² rigid (552 m² total)

All coefficients below are derived from these specifications.

Overview

The SWOPP3 model exposes two scalar functions:

Function	Description
predict_no_wps(tws, twa, swh, mwa, v)	Power without Wind Propulsion System
predict_with_wps(tws, twa, swh, mwa, v)	Power with Wind Propulsion System (sails)

Both return propulsion power in kW and accept:

Parameter	Symbol	Unit	Range
True wind speed	$\text{tws}$	m/s	$[0, 30]$
True wind angle	$\text{twa}$	deg	$[0, 180]$ (symmetric)
Significant wave height	$\text{swh}$	m	$[0, 10]$
Mean wave angle	$\text{mwa}$	deg	$[0, 180]$ (symmetric)
Ship speed	$v$	m/s	$[0, 14.5]$

predict_no_wps — Closed-Form Model

The total power is the sum of three perfectly additive components, clamped at zero:

$$ P_{\text{no_wps}} = \max!\Big(0,; P_{\text{hull}} + P_{\text{wind}} + P_{\text{wave}}\Big) $$

Hull Resistance

Pure cubic dependence on ship speed, independent of environment:

$$ P_{\text{hull}} = K_h \cdot v^3 $$

$$ K_h = \frac{969}{226} \approx 4.28761 $$

Physical interpretation: Hydrodynamic drag in calm water. The cubic law follows from drag force $\propto v^2$ multiplied by speed to get power.

$$ K_h = \frac{1}{2} \cdot \rho_{\text{water}} \cdot S \cdot C_T \cdot \frac{1}{1000} $$

With $\rho_{\text{water}} = 1025 ;\text{kg/m}^3$ and $S \approx 2200 ;\text{m}^2$:

$$ C_T = \frac{K_h}{\frac{1}{2} \cdot 1025 \cdot 2200 \cdot 10^{-3}} \approx 0.0038 $$

This total resistance coefficient is within the typical range $0.002\text{--}0.005$ for cargo ships at service speed.

Literature validation: The cubic dependence is consistent with the following formula found in naval papers:

$$ P_{\text{base}} = \frac{\Delta^{2/3} \cdot v^3}{3.7 \left( \sqrt{L} + 75 / v \right) } $$

where $\Delta$ is displacement and $L$ is the length of the vessel.

References:

• Molland, A.F., Turnock, S.R., Hudson, D.A. (2017). Ship Resistance and Propulsion, 2nd ed. Cambridge University Press. Chapter 3. doi:10.1017/9781316494196
• Holtrop, J., Mennen, G.G.J. (1982). "An approximate power prediction method." International Shipbuilding Progress, 29(335), 166–170. doi:10.3233/ISP-1982-2933501

Aerodynamic (Wind) Resistance

Depends on the apparent wind seen by the ship:

$$ P_{\text{wind}} = K_a \cdot v \cdot \big(V_R \cdot u_x - v^2\big) $$

where the apparent wind components are:

$$ u_x = \text{tws} \cdot \cos!\left(\text{twa} \cdot \frac{\pi}{180}\right) + v \qquad u_y = \text{tws} \cdot \sin!\left(\text{twa} \cdot \frac{\pi}{180}\right) $$

$$ V_R = \sqrt{u_x^2 + u_y^2} $$

$$ K_a = \frac{49}{320} = 0.153125 $$

Physical interpretation:

$$ K_a = \frac{1}{2} \cdot \rho_{\text{air}} \cdot C_D \cdot A_T \cdot \frac{1}{1000} $$

where $\rho_{\text{air}} = 1.225 ;\text{kg/m}^3$. The factor $1/1000$ converts W to kW. Solving for the drag-area product:

$$ C_D \cdot A_T = \frac{K_a}{\frac{1}{2} \cdot 1.225 \cdot 10^{-3}} = 250 ;\text{m}^2 $$

With $C_D \approx 0.7$ (typical for cargo ship superstructure), the implied frontal area is $A_T \approx 357 ;\text{m}^2$. For an 88 m vessel with $\sim 15 ;\text{m}$ beam and $\sim 20 ;\text{m}$ air draught, this is physically consistent ($15 \times 20 = 300 ;\text{m}^2$ plus rigging and wingsail structure).

Note: At large TWA with strong tailwinds, $P_{\text{wind}}$ becomes negative (wind assists the ship), which can drive total power to zero (clamped).

Literature validation: The formulation is consistent with the standard aerodynamic wind load model used in naval architecture:

$$ P_{\text{wind}} = \frac{1}{2} \cdot C_X \cdot \rho_{\text{air}} \cdot A_x \cdot \cos(\phi) \cdot v_{\text{wind}}^2 \cdot v $$

which follows from the classical drag expression

$$ F_{\text{wind}} = \frac{1}{2} , \rho_{\text{air}} , C_X , A_x , v_{\text{wind}}^2 $$

with power obtained as $P = F \cdot v$. Here $C_X$ is the longitudinal aerodynamic force coefficient, $A_x$ is the projected frontal area, and $\phi$ accounts for the wind attack angle.

References:

• Blendermann, W. (1994). "Parameter identification of wind loads on ships." Journal of Wind Engineering and Industrial Aerodynamics, 51(3), 339–351. doi:10.1016/0167-6105(94)90067-1
• Fujiwara, T., Ueno, M., Nimura, T. (1998). "Estimation of wind forces and moments acting on ships." Journal of the Society of Naval Architects of Japan, 183, 77–90. doi:10.2534/jjasnaoe1968.1998.77
• ITTC (2014). "Recommended Procedures and Guidelines: Speed and Power Trials." 7.5-04-01-01.1. ittc.info

Wave-Added Resistance

Factorizes cleanly into three independent terms:

$$ P_{\text{wave}} = A_w \cdot \text{swh}^2 \cdot v^{3/2} \cdot \exp!\Big(-K_w \cdot |\theta_{\text{mwa}}|^3\Big) $$

where $\theta_{\text{mwa}} = \text{mwa} \cdot \pi / 180$ is the wave angle in radians.

$$ A_w \approx 11.1395 \qquad K_w = \frac{125}{432} = \frac{5^3}{2^4 \cdot 3^3} \approx 0.28935 $$

Key properties:

• Quadratic in SWH ($\propto \text{swh}^2$) — exact
• Speed exponent is exactly $3/2$ ($\propto v^{1.5}$)
• Directional factor $\exp(-K_w |\theta|^3)$ decays from 1.0 at head seas ($\text{mwa}=0°$) to $\approx 0.00013$ at following seas ($\text{mwa}=180°$)

Literature validation: The three standard semi-empirical frameworks for added resistance in waves all share the same dimensional structure:

• Gerritsma & Beukelman (1972) derive added resistance from strip theory as an integral of relative wave-induced motions along the hull. The result scales as $R_{\text{aw}} \propto \rho g B^2 H^2 / L$, where $B$ is beam and $L$ is length — i.e. quadratic in wave height with a geometric hull-shape prefactor.

• Faltinsen et al. (1980) extend this to short waves using an asymptotic diffraction formulation. Their result also gives $R_{\text{aw}} \propto H^2$ and adds an explicit heading dependence through the encounter angle $\beta$.

• ITTC (2014) recommends the simplified Stawave-2 formula: $$ R_{\text{aw}} = \frac{1}{16} , \rho , g , H^2 , B , \sqrt{B/L} $$ which is speed-independent (resistance does not depend on $v$), so power grows as $P = R_{\text{aw}} \cdot v \propto H^2 \cdot v^1$.

All three frameworks agree on the $H^2$ dependence (a direct consequence of linear wave theory, where wave energy $\propto H^2$). They differ on the speed dependence of the resistance:

Framework	$R_{\text{aw}}$ speed dependence	$P_{\text{wave}}$ speed dependence
ITTC Stawave-2	$\propto v^0$	$\propto H^2 \cdot v^1$
Gerritsma-Beukelman (motions-based)	$\propto v^{0.5\text{--}1}$ (varies with encounter frequency)	$\propto H^2 \cdot v^{1.5\text{--}2}$
Experimental regressions	$\propto v^{0.5\text{--}1.5}$	$\propto H^2 \cdot v^{1.5\text{--}2.5}$

The SWOPP3 model uses $P_{\text{wave}} \propto v^{3/2}$, which implies a resistance that grows as $R_{\text{aw}} \propto v^{1/2}$. This sits between the speed-independent ITTC approximation and the motions-based strip theory results — a physically natural compromise. The $3/2$ exponent is best understood as a regression fit that captures the average speed sensitivity across the operating envelope, rather than a first-principles derivation.

Strong directional decay from head to following seas is also consistent with all three frameworks: added resistance is maximum in head seas and negligible in following seas, where the ship rides with the wave crests.

References:

• Gerritsma, J., Beukelman, W. (1972). "Analysis of the resistance increase in waves of a fast cargo ship." International Shipbuilding Progress, 19(217), 285–293. doi:10.3233/ISP-1972-1921701
• Salvesen, N. (1978). "Added resistance of ships in waves." Journal of Hydronautics, 12(1), 24–34. doi:10.2514/3.63110
• Faltinsen, O.M., Minsaas, K.J., Liapis, N., Skjørdal, S.O. (1980). "Prediction of resistance and propulsion of a ship in a seaway." Proc. 13th Symposium on Naval Hydrodynamics, Tokyo, 505–529. (Conference proceedings, no DOI available)

Accuracy

Tested against the reference binary on 10,000 random inputs:

Metric	Value
Mean absolute error	0.008 kW
p99 absolute error	0.064 kW
Max absolute error	0.114 kW
Max relative error	0.031%

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predict_with_wps — Sail-Assisted Model (Closed-Form)

$$ P_{\text{with_wps}} = \max!\Big(0,; P_{\text{hull}} + P_{\text{wind}} + P_{\text{wave}} - P_{\text{sail}}\Big) $$

where hull, wind, and wave terms are identical to predict_no_wps.

Sail Power — Closed-Form

The sail power saving factorizes exactly as:

$$ P_{\text{sail}} = C(\text{AWA}) \cdot V_R^2 \cdot v $$

where $V_R^2 = u_x^2 + u_y^2$ is the squared apparent wind speed and $\text{AWA}$ is the apparent wind angle:

$$ \text{AWA} = \arctan2!\big(|u_y|,; u_x\big) \cdot \frac{180}{\pi} $$

so $\text{AWA}$ is in degrees.

Sail polar coefficient $C(\text{AWA})$:

$$ C(\text{AWA}) = \begin{cases} 0 & \text{if } \text{AWA} < 10° \[4pt] K_s \cdot \sin\alpha \cdot \Big(1 + \dfrac{3}{20}\sin^2\alpha\Big) & \text{if } \text{AWA} \geq 10° \end{cases} $$

where $\alpha = (\text{AWA} - 10°) \cdot \pi / 180$ (converted to radians) and:

$$ K_s = 0.85903125 $$

Derivation of $K_s$: The value 0.85903125 was identified by isolating the sail contribution from the binary. Setting $\text{AWA}$ to a known angle where $\sin\alpha(1 + \frac{3}{20}\sin^2\alpha)$ evaluates to a clean value and solving for $K_s = P_{\text{sail}} / (V_R^2 \cdot v \cdot C(\alpha))$ yields the decimal 0.85903125 exactly. This is a terminating decimal:

$$ 0.85903125 = \frac{85903125}{10^8} = \frac{27489}{32000} = \frac{3 \cdot 7^2 \cdot 11 \cdot 17}{2^8 \cdot 5^3} $$

In other words, $K_s = 27489/32000$ — a ratio with a power-of-ten denominator, confirming it was likely chosen analytically rather than fitted numerically.

Key Properties

1. Wave-independent: $P_{\text{sail}}$ depends only on $(\text{tws}, \text{twa}, v)$ — no dependence on SWH or MWA.

2. Operates in apparent wind coordinates: The model naturally uses $\text{AWA}$ and $V_R$, not the true wind quantities directly.

3. 10° dead zone: The sail produces no power when $\text{AWA} &lt; 10°$ (too close to head wind). This cutoff is exact in apparent wind angle.

4. Peak at AWA ≈ 100°: $C(100°) = K_s \cdot 23/20 \approx 0.9879$ — the peak of the sail thrust polar occurs at beam-reach conditions.

5. Always non-negative: $P_{\text{sail}} \geq 0$ (sails only help).

6. Symmetric: $P_{\text{sail}}(\text{twa}) = P_{\text{sail}}(-\text{twa})$.

7. Physical interpretation: $K_s \cdot \sin\alpha$ is the primary lift-based thrust; the $\frac{3}{20}\sin^2\alpha$ term is a quadratic drag/lift correction that enhances thrust at large angles of attack.

   $$ K_s = \frac{1}{2} \cdot \rho_{\text{air}} \cdot C_L \cdot A_{\text{sail}} \cdot \frac{1}{1000} $$

   With total sail area $A_{\text{sail}} = 4 \times 138 = 552 ;\text{m}^2$:

   $$ C_L = \frac{K_s}{\frac{1}{2} \cdot 1.225 \cdot 552 \cdot 10^{-3}} \approx 2.54 $$

   This effective lift coefficient is within the typical range $2\text{--}4$ for rigid wingsails.

Accuracy

The closed-form sail model is exact to machine precision ($&lt; 10^{-13}$) against the reference. The only residual error in the full model comes from the $A_w$ constant in the wave term.

Tested against the reference binary on 50,000 random inputs:

Metric	Value
Mean absolute error	0.004 kW
p99 absolute error	0.030 kW
Max absolute error	0.050 kW
Errors > 0.1 kW	0 / 50,000

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Summary of Constants

Constant	Symbol	Value	Exact
Hull coefficient	$K_h$	4.28761…	$969/226$
Air drag coefficient	$K_a$	0.153125	$49/320$
Wave amplitude	$A_w$	11.1395	fitted
Wave directional decay	$K_w$	0.28935…	$125/432 = 5^3/(2^4 \cdot 3^3)$
Sail thrust coefficient	$K_s$	0.85903125	$27489/32000$
Sail dead zone angle	—	10°	exact
Sail quadratic correction	—	3/20 = 0.15	exact

Physical Coefficients (88 m Cargo Vessel)

Coefficient	Symbol	Derived Value	Typical Range	Source
Total resistance	$C_T$	0.0038	0.002–0.005	$K_h / (\frac{1}{2} \rho_w S / 1000)$, $S \approx 2200;\text{m}^2$
Wind drag × area	$C_D A_T$	250 m²	—	$K_a / (\frac{1}{2} \rho_a / 1000)$
Wind drag	$C_D$	~0.7	0.6–0.9	Assuming $A_T \approx 357;\text{m}^2$
Wingsail lift	$C_L$	2.54	2–4	$K_s / (\frac{1}{2} \rho_a \cdot 552 / 1000)$

Where $\rho_w = 1025 ;\text{kg/m}^3$, $\rho_a = 1.225 ;\text{kg/m}^3$.

Wave Dominance

At typical service speed ($v = 8;\text{kn}$), wave-added resistance exceeds hull resistance at surprisingly low wave heights:

SWH (m)	$P_{\text{wave}}$ / $P_{\text{hull}}$	$P_{\text{wave}}$ share
1.0	0.31	24%
1.8	1.00	50% (crossover)
3.0	2.78	74%
4.0	4.98	83%

(Head seas, $\text{mwa} = 0°$.) This is consistent with the SWOPP3 binary itself — wave dominance is a genuine physical property of the model, not a modelling artifact.

Clamping Rule

Both functions clamp the result at zero:

$$ P = \max(0, P_{\text{raw}}) $$

This occurs when strong tailwinds ($P_{\text{wind}} &lt; 0$) or large sail savings ($P_{\text{sail}}$) exceed hull + wave resistance.