PWR-HYBRID-3/docs/model_cheatsheet.md

PWR Plant Model Cheatsheet

Quick reference for the 10-state PKE + 3-node thermal-hydraulics model used throughout this demo.

Source of truth: code/src/pke_params.jl (constants + steady state), code/src/pke_th_rhs.jl (dynamics), code/src/pke_th_rhs_pj.jl (prompt-jump variant), code/controllers/controllers.jl (control laws).

All values in SI internally; printed/plotted in °F.

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State vector

x = [n, C₁, C₂, C₃, C₄, C₅, C₆, T_f, T_c, T_cold] (10 states)

Symbol	Index	Meaning	Units
n	1	Normalized neutron population (n=1 ⇔ full power)	—
Cᵢ	2–7	Delayed-neutron precursor concentrations (6 groups)	—
T_f	8	Fuel-pellet bulk temperature	°C
T_c	9	Average coolant temperature in core (= T_avg)	°C
T_cold	10	Cold-leg coolant temperature	°C

Derived (not states):

• T_hot = 2·T_c − T_cold (linear coolant profile assumption)
• P = P₀·n (thermal power)

Inputs:

• u = control input (rod-induced reactivity), units of Δk/k
• Q_sg(t) = bounded disturbance (steam-generator heat removal), W

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Reactivity (the coupling between neutronics and thermal)

\rho(x, u) \;=\; u \;+\; \alpha_f\,(T_f - T_{f,0}) \;+\; \alpha_c\,(T_c - T_{c,0})

Three contributions:

• u — externally controlled (rod motion)
• α_f·(T_f − T_f0) — Doppler feedback (negative; fuel heats → more U-238 absorption → less reactivity)
• α_c·(T_c − T_c0) — moderator-density feedback (negative; coolant heats → less moderation → less reactivity)

Both feedback coefficients are negative — the plant is self-stabilizing.

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ODE system (full 10-state)

Neutronics (Point-Kinetic Equations)

\frac{dn}{dt} \;=\; \frac{\rho - \beta}{\Lambda}\,n \;+\; \sum_{i=1}^{6} \lambda_i C_i \frac{dC_i}{dt} \;=\; \frac{\beta_i}{\Lambda}\,n \;-\; \lambda_i C_i \qquad (i = 1, \ldots, 6)

The first equation has a stiff coefficient 1/Λ ≈ 10⁴ s⁻¹ — this is what makes the full plant unverifiable by direct nonlinear reach.

Thermal-hydraulics (3 lumped nodes)

\frac{dT_f}{dt} \;=\; \frac{P_0\,n \;-\; hA\,(T_f - T_c)}{M_f\,c_f} \frac{dT_c}{dt} \;=\; \frac{hA\,(T_f - T_c) \;-\; 2\,W\,c_c\,(T_c - T_{\text{cold}})}{M_c\,c_c} \frac{dT_{\text{cold}}}{dt} \;=\; \frac{2\,W\,c_c\,(T_c - T_{\text{cold}}) \;-\; Q_{sg}}{M_{sg}\,c_c}

Physical chain: fission heat → fuel → coolant → cold leg → SG → back to cold leg.

The factor of 2 in 2·W·c_c·(T_c − T_cold) comes from the linear-profile assumption: the enthalpy carried by flow at avg T_c from inlet T_cold to outlet T_hot = 2T_c − T_cold is W·c_c·(T_hot − T_cold) = 2·W·c_c·(T_c − T_cold).

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Prompt-jump (PJ) reduction

Set dn/dt = 0 and solve algebraically for n:

n(x) \;=\; \frac{\Lambda \sum_i \lambda_i C_i}{\beta - \rho(x, u)}

State drops 10 → 9 (drop n; reconstruct it from C and ρ). Stiffness gone. Validity condition: β − ρ > 0 with margin (encoded as the prompt_critical_margin_* halfspace per mode).

Tikhonov bound: |x(t) − x_PJ(t)| ≤ C·Λ = O(10⁻⁴) — characterized residual error vs the 10-state plant.

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Constants

Neutronics (6-group point-kinetic)

Symbol	Value	Meaning
Λ	1×10⁻⁴ s	Prompt-neutron generation time
β = Σβᵢ	0.006502	Total delayed-neutron fraction (sum of 6 groups)
β₁..β₆	[0.000215, 0.001424, 0.001274, 0.002568, 0.000748, 0.000273]	Per-group delayed fractions
λ₁..λ₆	[0.0124, 0.0305, 0.111, 0.301, 1.14, 3.01] s⁻¹	Precursor decay constants

(Standard 6-group U-235 thermal-fission values from Keepin.)

Thermal-hydraulic

Symbol	Value	Meaning
P₀	1×10⁹ W	Rated thermal power (1 GWth)
M_f	50 000 kg	Fuel mass (lumped)
c_f	300 J/(kg·K)	Fuel specific heat
M_c	20 000 kg	Core coolant mass
c_c	5 450 J/(kg·K)	Coolant specific heat
hA	5×10⁷ W/K	Fuel-to-coolant heat-transfer coefficient
W	5 000 kg/s	Primary coolant mass flow rate
M_sg	30 000 kg	Steam-generator coolant inventory

Reactivity feedback coefficients (linearized about full-power op point)

Symbol	Value	Meaning
α_f	−2.5×10⁻⁵ /°C	Fuel/Doppler temperature coefficient
α_c	−1×10⁻⁴ /°C	Moderator/coolant temperature coefficient

These are linear slopes about the full-power operating point. Trust range: ~±50 °C around T_f0/T_c0. Cold-shutdown extrapolation breaks.

Derived steady-state (full-power equilibrium)

Single free parameter: T_cold0 = 290 °C (full-power cold-leg temp). Everything else falls out of energy balance:

Quantity	Derivation	Value
ΔT_core	P₀ / (W·c_c)	36.7 °C
T_hot0	T_cold0 + ΔT_core	326.7 °C
T_c0 (= T_avg0)	(T_hot0 + T_cold0) / 2	308.35 °C
T_f0	T_c0 + P₀/hA	328.35 °C
n₀	(definition)	1.0
C_i0	βᵢ·n₀ / (λᵢ·Λ)	(from dC/dt = 0)

Hot-standby reference

Quantity	Derivation	Value
T_standby	T_c0 − 33.33 °C (= T_c0 − 60 °F)	275.02 °C

This is the canonical "hot standby" operating point — coolant warm but power below criticality. Defined as a 60 °F offset below full-power T_avg.

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Per-mode controllers

Mode	Control law	Constants	Notes
q_shutdown	u = −5β (constant rod)	—	Open-loop rod-held
q_heatup	u = K_p·(T_ref(t) − T_c) (FL/proportional)	K_p = 1×10⁻⁴	T_ref ramps from T_standby at 28 °C/hr (tech-spec heatup limit)
q_operation (P)	u = K_p·(T_avg_ref − T_avg)	K_p ≈ 1×10⁻⁴	Simple proportional T_avg tracker
q_operation (LQR)	u = −K_LQR·δx	gain solved from Riccati	Cached in factory closure
q_scram	u = −8β (constant rod, max insertion)	—	Open-loop rod-held; full negative reactivity

Heatup ramp rate: dT_ref/dt = 28/3600 ≈ 0.00778 °C/s (28 °C/hr — US PWR tech-spec maximum heatup rate, set by vessel thermal-stress limits).

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Key predicates / halfspaces (used by reach analysis)

From reachability/predicates.json:

Predicate	Concretization	What it discharges
t_avg_above_min	T_c ≥ T_standby + 5.556 °C (= 280.58 °C)	Shutdown→heatup transition trigger
t_avg_in_range	|T_c − T_c0| ≤ 2.78 °C (= [305.57, 311.13])	Heatup→operation transition trigger
p_above_crit	n ≥ 1×10⁻⁴	"Reactor at-or-above critical" (also part of heatup→operation)
fuel_centerline	T_f ≤ 1200 °C	UO₂ melt prevention
t_avg_high_trip	T_c ≤ 320 °C	Reactor trip limit
t_avg_low_trip	T_c ≥ 280 °C	Reactor trip limit
n_high_trip	n ≤ 1.15	High-flux trip (118% nominal)
cold_leg_subcooled	T_cold ≤ T_cold0 + 15 = 305 °C	Subcooling margin
heatup_rate_upper	0.4587·T_f − 0.9587·T_c + 0.5·T_cold ≤ 0.01389 °C/s	Coolant heatup ≤ 50 °C/hr (28 + overshoot)
heatup_rate_lower	(mirror, lower bound)	Cooldown ≤ 50 °C/hr
prompt_critical_margin_*	β − ρ ≥ δ (per-mode form)	PJ reduction validity
shutdown_margin	α_f·T_f + α_c·T_c ≤ 0.00297	Scram success: rho ≤ −0.01 (1% Δk/k)

The dT_c/dt halfspace coefficients above come from differentiating the T_c ODE: a_f = hA/(M_c·c_c) = 0.4587 /s, a_c = −(hA + 2W·c_c)/(M_c·c_c) = −0.9587 /s, a_cold = 2W·c_c/(M_c·c_c) = 0.5 /s. Sums to zero by equilibrium.

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Per-mode obligations

Mode	Kind	Obligation
q_shutdown	equilibrium	Stay in X_safe forever; transition out when t_avg_above_min becomes true
q_heatup	transition	From X_entry, reach X_exit within [T_min=7714, T_max=18000] s, maintain inv1_holds
q_operation	equilibrium	Stay in X_safe forever under bounded Q_sg
q_scram	transition	From any state, drive to shutdown_margin within T_max=60 s, maintain bounded T

inv1_holds (heatup safety invariant) = conjunction of fuel_centerline, cold_leg_subcooled, heatup_rate_upper, heatup_rate_lower, prompt_critical_margin_heatup.

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Sanity numbers (to memorize before the talk)

Quantity	Value
Full-power n	1.0
Full-power T_avg	308 °C ≈ 587 °F
Hot-standby T_avg	275 °C ≈ 527 °F
Heatup span (standby → operation)	33 °C ≈ 60 °F
Tech-spec heatup rate	28 °C/hr
Nominal heatup time	33 / 28 = 1.19 hr ≈ 71 min
T_min (heatup obligation)	7 714 s = 2 hr 8 min
T_max (heatup obligation)	18 000 s = 5 hr
Stiffness ratio (full plant)	~10⁵ (Λ⁻¹ ÷ thermal time-constant)
Scram rod worth	−8β = −0.052 Δk/k (~5% subcritical)
Shutdown-margin requirement	|ρ| ≥ 0.01 = 1% Δk/k