Sensor–state dependencies in ADM1da¶

This document is step 2 of the observability analysis: the concrete reconciliation of the published literature (step 1) with our ADM1da implementation in pyadm1.core.adm1.

Goal: for each measurement channel, document which of the 41 ADM1da states appear directly in the measurement equation and which become observable only indirectly through the model's dynamics. From this follows the rationale for which states a UKF can realistically estimate when only certain sensors are available.

Measurements are split by availability — an online sensor delivers innovation every second, lab analytics every few days, research-grade methods maybe once per year.

Methodology¶

A state \(x_i\) is observable from a measurement \(y = h(x)\) only if either

directly \(x_i\) appears explicitly in \(h(x)\) (nonzero derivative in
the measurement model), or
indirectly \(x_i\) appears in the right-hand side \(\dot x_j = f_j(x)\)
of some directly observed \(x_j\) — then \(\dot y\) carries information about \(x_i\). The mechanism iterates: after \(k\) steps, \(x_i\) appears in \(y^{(k)}\) if a chain of length \(k\) exists through the Jacobian (Lie derivatives, see Hellmann et al. 2023).

In practice:

D = direct — immediate observability in a single update step.
I₁ = 1-step indirect — observability after at least one time step
with dynamics in between.
I₂+ = k-step indirect — reconstructible only with very accurate
model dynamics and a long observation window; in practice often inseparable from model errors.
C = correlative — no direct model expression, but empirically
coupled (e.g. conductivity ↔ ionic strength).

The tables below cite code locations in pyadm1/core/adm1.py (line numbers refer to v0.3.4).

Notation and symbols¶

Glossary of the most important quantities so the equations and tables below stay readable. The full 41-state index map is in Theory → ADM1da model.

State classes¶

Prefix	Meaning	Typical unit
\(S_\bullet\)	dissolved species (solute) in the liquid	kg COD/m³ or kmol/m³
\(X_\bullet\)	particulate species (biomass, solid pools)	kg COD/m³
\(p_{\text{gas},\bullet}\)	partial pressure of a gas-phase species	bar
\(p_{\text{total}}\) (= pTOTAL)	total gas pressure in the headspace, state index 40	bar

Subscripts (compound classes)¶

Code	Meaning
su	sugars (monosaccharides)
aa	amino acids
fa	long-chain fatty acids (LCFA)
va	valeric acid / valerate (C5)
bu	butyric acid / butyrate (C4)
pro	propionic acid / propionate (C3)
ac	acetic acid / acetate (C2)
h2	hydrogen
ch4	methane
co2	inorganic carbon (= S_IC)
nh4	ammonium nitrogen (= S_IN)
nh3	free ammonia
hco3	bicarbonate HCO₃⁻
_ion (suffix)	dissociated ionic form of a VFA (e.g. \(S_{ac\_ion}\) = acetate anion)
cation, anion	sum of all strong cations / anions [kmol/m³]
I	inert (dissolved \(S_I\) or particulate \(X_I\))
ch / pr / li	carbohydrates / proteins / lipids
PS / PF / S	pool class: slow-disintegrating / fast-disintegrating / hydrolysable
c4	C4-acid degraders (valerate / butyrate utilizers)
bac	bacterial biomass (generic)

Observable quantities (outputs / measurement models)¶

Symbol	Meaning
\(Q_{\text{gas}}\)	total biogas volume flow [m³/d]
\(Q_{\text{CH4}}\), \(Q_{\text{CO2}}\)	methane / CO₂ volume flow [m³/d]
\(P_{\text{gas}}\)	sum of dry partial pressures \(p_{\text{gas,h2}} + p_{\text{gas,ch4}} + p_{\text{gas,co2}}\) [bar]
\(p_{\text{total}}\)	total gas pressure as an integrated model state (index 40)
\(p_{\text{gas,h2o}}\)	water-vapour saturation pressure — a temperature-dependent parameter, not a state
pH	\(-\log_{10}(S_H)\) with \(S_H\) from the Newton solution of the charge balance
VFA	sum of volatile fatty acids as HAc equivalent [g HAc/L]
TAC	total alkalinity (titrated to pH 5) [g CaCO₃/L]
FOS/TAC	ratio of VFA to TAC — Nordmann early-warning indicator
\(Q_{\text{solid}}\), \(Q_{\text{liquid}}\)	augmented substrate feed rates [m³/d] — not ADM1 states but part of the extended state vector

Reaction rates \(\rho\) (in the code: `Rho_*`)¶

Symbol	Meaning
\(\rho_{\text{dis,PS}}\), \(\rho_{\text{dis,PF}}\)	disintegration rates (\(X_{PS} \to X_S\), \(X_{PF} \to X_S\))
\(\rho_{\text{hyd,ch/pr/li}}\)	hydrolysis rates (\(X_S \to S\))
\(\rho_{\text{su}}, \rho_{\text{aa}}, \rho_{\text{fa}}, \rho_{\text{c4}}, \rho_{\text{pro}}, \rho_{\text{ac}}, \rho_{\text{h2}}\)	substrate uptake rates (Monod kinetics × biomass × inhibition)
\(\rho_{\text{dec,}\bullet}\)	biomass decay rates
\(\rho_{A,\bullet}\)	acid-base reaction rates (very fast, \(k_{AB} = 10^8\))
\(\rho_{T,\text{h2}}, \rho_{T,\text{ch4}}, \rho_{T,\text{co2}}\)	gas-liquid transfer rates (Henry's law)
\(\rho_{T,11}\)	bulk gas outflow from the headspace

Inhibition factors \(I\) (values in [0, 1])¶

Symbol	Meaning
\(I_{\text{pH,aa}}, I_{\text{pH,ac}}, I_{\text{pH,h2}}\)	pH inhibition (Hill type, exponent 1-3 per pool)
\(I_{\text{IN}}\)	N limitation: \(S_{IN}/(K_{S,IN}+S_{IN})\)
\(I_{\text{h2,fa/c4/pro}}\)	H₂ inhibition for LCFA, C4, propionate degraders
\(I_{\text{nh3}}, I_{\text{nh3,pro}}\)	NH₃ inhibition for acetoclasts and propionate degraders
\(I_{\text{HAc}}, I_{\text{HPr}}\)	inhibition by undissociated acetic / propionic acid
\(I_{\text{co2,h2}}\)	CO₂ inhibition of the H₂-producing pathways

Key parameters¶

Symbol	Meaning
\(k_{m,\bullet}\), \(K_{S,\bullet}\)	maximum Monod rate and half-saturation constant
\(k_{\text{dec,}\bullet}\)	decay rate per biomass pool [d⁻¹]
\(k_{\text{dis,PS/PF}}\), \(k_{\text{hyd,ch/pr/li}}\)	disintegration / hydrolysis rates [d⁻¹]
\(Y_\bullet\)	biomass yield per substrate COD [-]
\(K_{a,\bullet}\)	acid dissociation constants
\(K_{H,\text{ch4/co2/h2}}\)	Henry constants (gas solubility)
\(K_w\)	ionic product of water
\(k_{L}a\)	gas-liquid transfer coefficient [d⁻¹]
\(k_p\)	gas-outlet flow coefficient (pipe flow)
\(V_{\text{liq}}\), \(V_{\text{gas}}\)	liquid / gas-phase volume [m³]
\(T_{\text{ad}}\)	operating temperature (adiabatic) [K]
\(R\), \(RT\)	gas constant / product \(R \cdot T_{\text{ad}}\)
\(M_{\text{HAc}}\)	60 g/mol (acetate molar mass) for VFA conversion
64, 112, 160, 208	COD equivalents for acetate / propionate / butyrate / valerate [g COD/mol]
\(f_{\bullet}\)	stoichiometric product fractions (e.g. \(f_{\text{ac,su}} = 0.41\))
\(N_{\text{bac}}, N_{\text{aa}}, N_I\)	nitrogen contents per biomass / amino-acid / inert pool [kmol N/kg COD]
\(D_{\text{in}}, D_{\text{out}}\)	dilution rates \(Q/V_{\text{liq}}\) [d⁻¹]
\(S_H\)	\(H^+\) concentration from the Newton solution of the charge balance

Measurement-model equations (from the code)¶

Gas volume flows¶

From ADM1.calc_gas:

# L. 451-454
q_gas = max(
    k_p * (p_total_wet - p_ext) / (RT/1000 * NQ) * V_liq,
    0.0,
)
# L. 456-460
p_gas    = pi_Sh2 + pi_Sch4 + pi_Sco2          # = p_gas_h2 + p_gas_ch4 + p_gas_co2
q_ch4    = q_gas * pi_Sch4 / (p_gas + p_gas_h2o)
q_co2    = q_gas * pi_Sco2 / (p_gas + p_gas_h2o)

Observable	Directly depends on	Idx
\(Q_{\text{gas}}\)	pTOTAL	40
\(P_{\text{gas}}\)	\(p_{\text{gas\_h2}}\), \(p_{\text{gas\_ch4}}\), \(p_{\text{gas\_co2}}\)	37, 38, 39
\(Q_{\text{CH4}}\)	pTOTAL, \(p_{\text{gas\_h2}}\), \(p_{\text{gas\_ch4}}\), \(p_{\text{gas\_co2}}\)	37, 38, 39, 40
\(Q_{\text{CO2}}\)	pTOTAL, \(p_{\text{gas\_h2}}\), \(p_{\text{gas\_ch4}}\), \(p_{\text{gas\_co2}}\)	37, 38, 39, 40

pH¶

From ADM1._calc_ph:

# L. 1046-1048
vfa_anions = S_ac_ion/64 + S_pro_ion/112 + S_bu_ion/160 + S_va_ion/208
fixed      = S_cation - S_anion + (S_nh4 - S_nh3) - S_hco3 - vfa_anions
# Newton: f(S_H) = fixed + S_H − K_w/S_H = 0
pH = -log10(S_H)

Observable	Directly depends on	Idx
pH	\(S_{nh4}, S_{nh3}, S_{hco3}, S_{ac\_ion}, S_{pro\_ion}, S_{bu\_ion}, S_{va\_ion}, S_{cation}, S_{anion}\)	10, 36, 35, 34, 33, 32, 31, 29, 30

VFA and TAC¶

From Digester._compute_indicators:

# L. 362-367  VFA in g HAc-eq / m³
vfa = M_HAc * (S_ac/64 + S_pro/112 + S_bu/160 + S_va/208)

# L. 389-399  TAC in g CaCO3 / m³ (endpoint pH 5)
# combines: free NH3/NH4, HCO3/CO2, all VFA ions, S_cation/S_anion.

Observable	Directly depends on	Idx
VFA sum	\(S_{va}, S_{bu}, S_{pro}, S_{ac}\)	3, 4, 5, 6
VFA individual (HPLC)	per acid individually	3, 4, 5 or 6
TAC	\(S_{va}, S_{bu}, S_{pro}, S_{ac}, S_{co2}, S_{nh4}, S_{cation}, S_{anion}, S_{va\_ion}, S_{bu\_ion}, S_{pro\_ion}, S_{ac\_ion}, S_{hco3}, S_{nh3}\)	3-6, 9, 10, 29-36
FOS/TAC	Union of VFA and TAC	ditto

A — Online measurements¶

Continuously available, second-to-minute innovation. This list covers the typical SCADA instrumentation of an agricultural biogas plant plus optional upgrades.

A.1 Standard SCADA (most plants)¶

Sensor	What is measured	Direct ADM1da-state link	Class
Q_gas (thermal / turbine flowmeter)	biogas volume flow [m³/d]	pTOTAL (40)	D
Gas storage level (% filled)	storage Δ over time → contribution to Q_gas	as Q_gas	reconstructive
Flare on/off	status; boundary for Q_gas balance	— (boundary)	—
p_head	headspace pressure [mbar/bar]	pTOTAL (40)	D
T_reactor (PT100 in digester)	fermenter temperature	— (boundary, scales kinetics via Arrhenius)	—
T_ambient, T_inlet	heat-balance boundary	—	—
Substrate load cell / pump counter	\(-\Delta W\) → Q_feed (solid or liquid)	\(Q_{solid}\) / \(Q_{liquid}\) (augmented)	D
CHP Q_ch4 consumption	methane consumption per CHP [m³/h]	pTOTAL + p_gas_ch4 (40, 38) via \(\eta\)	I₁
CHP P_el (electrical power)	electrical output [kW]	as Q_ch4, derived via \(\eta_{el}\)	I₁
Stirrer current / power	mixer — viscosity proxy	— (no ADM1 coupling)	—
Heating power	thermal output [kW]	— (boundary)	—

A.2 Online gas analytics (NDIR multi-gas sensor)¶

Sensor	What is measured	Direct ADM1da-state link	Class
CH₄ fraction in biogas (NDIR)	\(p_{gas,ch4}/p_{gas}\) [%]	\(p_{gas,ch4}\) (38)	D
CO₂ fraction in biogas (NDIR)	\(p_{gas,co2}/p_{gas}\) [%]	\(p_{gas,co2}\) (39)	D
H₂ fraction in biogas (TCD/electrochem)	\(p_{gas,h2}/p_{gas}\) [ppm-%]	\(p_{gas,h2}\) (37)	D
O₂ fraction in biogas	quality check (air leakage)	— (no ADM1 state)	—
H₂S fraction in biogas	hydrogen sulfide [ppm]	— (no ADM1 state; desulfurization)	—

A.3 Online liquid-phase (premium instrumentation)¶

Sensor	What is measured	Direct ADM1da-state link	Class
pH probe (glass electrode, ISFET)	liquid pH	9 charge-balance states (10, 29-36)	D
ORP / redox potential	reduction potential [mV]	\(S_{h2}\) (7), methanogenic activity	C
Conductivity (EC)	electrical conductivity [µS/cm]	\(S_{cation} + S_{anion}\) + all ion species	C
Online TAC (FOS titrator)	alkalinity [g CaCO₃/L]	as lab TAC, see below	D
Online VFA (HPLC online)	individual acids [g/L]	\(S_{va}, S_{bu}, S_{pro}, S_{ac}\) individually (3-6)	D
Online NH₄⁺ (ion-selective electrode)	ammonium [g/L]	\(S_{nh4}\) (10)	D
Level / V_liq	fill level	— (constant in current model)	—

B — Lab analytics¶

Periodic (daily, weekly, monthly). Delivers sample-and-hold values; the UKF uses them as gated observations at sporadic times.

B.1 Standard AD routine (typ. weekly)¶

Lab quantity	Method	Direct ADM1da-state link	Class
pH (lab)	glass electrode	as online pH	D
VFA (FOS titration Nordmann)	titrimetric	VFA sum: \(S_{va}, S_{bu}, S_{pro}, S_{ac}\) as sum	D-sum
TAC (alkalinity titration)	titrimetric	as above (TAC entry)	D
FOS/TAC	derived	ratio	D
TS (total solids)	drying oven	\(X_I\) directly, particulate pools as sum	I₁(Σ)
VS / oTS (loss on ignition)	ignition at 550 °C	analogous to TS, without mineral ash	I₁(Σ)
NH₄-N	indophenol or IC	\(S_{nh4}\) (10)	D
Conductivity (lab)	EC meter	\(S_{cation} + S_{anion}\) sum	C

B.2 Extended analytics (typ. weekly to monthly)¶

Lab quantity	Method	Direct ADM1da-state link	Class
Individual VFA (acetate, propionate, n-/iso-butyrate, n-/iso-valerate)	HPLC or GC	\(S_{ac}\) (6), \(S_{pro}\) (5), \(S_{bu}\) (4), \(S_{va}\) (3) each	D per acid
TKN (Total Kjeldahl Nitrogen)	Kjeldahl digestion	\(S_{nh4} + S_{nh3} + N\) content of all biomass and protein pools	D-sum over N
COD total	photometric	sum of all COD species	I₁(Σ)
COD dissolved (filtered 0.45 µm)	as above	only dissolved COD: \(S_{su, aa, fa, va, bu, pro, ac, h2, ch4, I}\) sum	I₁(Σ-dissolved)
COD particulate	COD_total − COD_dissolved	particulate pools sum	I₁(Σ-part.)
Cation inventory (Na⁺, K⁺, Ca²⁺, Mg²⁺)	IC or ICP-OES	\(S_{cation}\) (29)	D
Anion inventory (Cl⁻, SO₄²⁻, PO₄³⁻, NO₃⁻)	IC	\(S_{anion}\) (30)	D
BOD5	5-day respiration test	biodegradable COD fraction	I₂(Σ)

B.3 Substrate analytics (for s_in of the influent balance)¶

Lab quantity	What it delivers	Direct ADM1da-input link
Substrate TS, oTS, COD	per-substrate characterization	s_in[i] for particulate / inert pools per substrate slot
Substrate NH₄-N, TKN	nitrogen contribution	s_in[10], s_in[11]
Substrate disintegration fractions	from profiles or lab BMP test	f_ch_xc, f_pr_xc, f_li_xc, f_xi_xc, f_si_xc — feeds the calibration artifact, not state

Substrate analytics does not affect the filter state directly, but the influent vector \(s_{in}\) — i.e. it shifts the initial condition of the mass balance, not the state itself. A wrongly parametrized influent leads to chronic filter biases.

C — Research / specialized methods (for completeness)¶

Method	What is measured	Direct ADM1da-state link	Note
qPCR / 16S rRNA sequencing	biomass populations quantitatively	\(X_{su}, X_{aa}, X_{fa}, X_{c4}, X_{pro}, X_{ac}, X_{h2}\) (22-28) individually	expensive (~€100/sample), weeks lead time — rare in production
GC-MS for LCFA / trace VFA	individual LCFA species, trace acids	\(S_{fa}\) (2) and finer subscale	research
In-situ H₂ probe (headspace)	online \(p_{gas,h2}\)	\(p_{gas,h2}\) (37)	see A.2
Trace-element analytics (Fe, Co, Ni, Mo, Se)	important for kinetics	— (no ADM1 state, affects \(k_{m,*}\))	as calibration hint
H₂S dissolved, sulfate, sulfide	sulfur inhibition	— (not in ADM1da)	for Hill-extended models
BMP test (batch methane potential)	specific methane yield per substrate	\(f_{*}\) fractions, k_dis per substrate	input to calibration artifact

ODE coupling structure — who depends on whom¶

The right-hand side of every differential equation contains further states. The dependencies below are extracted from ADM_ODE:

Gas-phase ODEs (L. 914-918)¶

diff_p_h2  = Rho_T_h2  * RT/16 − p_gas_h2/pTOTAL  * Rho_T_11
diff_p_ch4 = Rho_T_ch4 * RT/64 − p_gas_ch4/pTOTAL * Rho_T_11
diff_p_co2 = Rho_T_co2 * RT    − p_gas_co2/pTOTAL * Rho_T_11
diff_pTOT  = RT/16 * Rho_T_h2 + RT/64 * Rho_T_ch4 + RT * Rho_T_co2 − Rho_T_11

with (L. 712-723):

Rho_T_h2  = k_L_a * (S_h2  - 16*p_gas_h2 /(RT*K_H_h2 )) * V_liq/V_gas
Rho_T_ch4 = k_L_a * (S_ch4 - 64*p_gas_ch4/(RT*K_H_ch4)) * V_liq/V_gas
Rho_T_co2 = k_L_a * (S_co2_free - p_gas_co2/(RT*K_H_co2)) * V_liq/V_gas
S_co2_free = max(S_co2 - S_hco3, 0)
Rho_T_11  = k_p * (pTOTAL + p_gas_h2o − p_ext) * V_liq/V_gas

→ A gas-flow measurement directly exposes within one time step: pTOTAL (D), \(p_{gas,h2/ch4/co2}\) (1-step indirect via \(\dot p_{TOT}\)), \(S_{h2}, S_{ch4}, S_{co2}, S_{hco3}\) (all via \(\rho_T\) terms).

Methane chain (S_ch4 ODE, L. 828-834)¶

diff_S_ch4 = D_in·s_in[8] - D_out·S_ch4 + (1-Y_ac)·Rho_ac + (1-Y_h2)·Rho_h2
             - V_gas/V_liq · Rho_T_ch4

with \(\rho_{ac} = k_{m,ac}\cdot S_{ac}/(K_{S,ac}+S_{ac})\cdot X_{ac}\cdot I_{ac}\) and \(\rho_{h2} = k_{m,h2}\cdot S_{h2}/(K_{S,h2}+S_{h2})\cdot X_{h2}\cdot I_{h2}\) (L. 686-687).

→ Additionally reachable from Q_CH4 after 2 steps: \(X_{ac}, X_{h2}, S_{ac}\), and all inhibition factors \(I_{ac}\) (= all ions + \(S_{nh3}\)).

Acetate chain (S_ac ODE, L. 805-815)¶

diff_S_ac = D_in·s_in[6] - D_out·S_ac
            + (1-Y_su)·f_ac_su  · Rho_su      # X_su, S_su
            + (1-Y_aa)·f_ac_aa  · Rho_aa      # X_aa, S_aa
            + (1-Y_fa)·0.7      · Rho_fa      # X_fa, S_fa
            + (1-Y_c4)·0.31     · Rho_c4_va   # X_c4, S_va
            + (1-Y_c4)·0.8      · Rho_c4_bu   # X_c4, S_bu
            + (1-Y_pro)·0.57    · Rho_pro     # X_pro, S_pro
            - Rho_ac

→ A VFA measurement (sees \(S_{ac}\) directly) after one step exposes: \(S_{su}, S_{aa}, S_{fa}, S_{va}, S_{bu}, S_{pro}\) and \(X_{su}, X_{aa}, X_{fa}, X_{c4}, X_{pro}, X_{ac}\).

Nitrogen chain (S_nh4 ODE, L. 840-853)¶

diff_S_nh4 = D_in·s_in[10] - D_out·S_nh4
             - Y_su·N_bac·Rho_su + (N_aa - Y_aa·N_bac)·Rho_aa
             - Y_fa·N_bac·Rho_fa - ...
             + (N_bac - f_pr_bac·N_aa - f_p_bac·N_I)·sum_decay
             + Rho_A_IN

→ NH4-N measurement additionally exposes: all biomass decay pools, \(S_{nh3}\) via \(\rho_{A,IN}\), and via \(S_H\) the entire pH algebra.

Particulate pools (L. 856-893)¶

Disintegration X_PS_ → X_S_ and hydrolysis X_S_ → S_su/aa/fa act on sums — the ch/pr/li split appears identically everywhere and is therefore not separable* with aggregated measurements (TS, VS, COD).

Master table: sensor → exposed states¶

Synthesis of direct + 1-step indirect dependencies, separated by availability. Symbols: D direct · I₁ 1-step · I₂+ multi-step · C correlative · Σ only separable as sum · (blank) no realistic path.

Online sensors¶

Dissolved components (0-11)¶

Sensor	S_va	S_bu	S_pro	S_ac	S_h2	S_ch4	S_co2	S_nh4
Q_gas					I₁	I₁	I₁
Q_ch4 (NDIR)				I₂	I₁	I₁	I₂	I₂
Q_co2 (NDIR)				I₂			I₁	I₂
H₂ gas (NDIR)					I₁		I₂
pH (online)								D
Conductivity								C
ORP					C
Online VFA (HPLC)	D	D	D	D
Online NH₄⁺ (ISE)								D
CHP Q_ch4				I₂	I₁	I₁	I₂

Biomass + augmented inputs¶

Sensor	X_su	X_aa	X_fa	X_c4	X_pro	X_ac	X_h2	Q_solid	Q_liquid
Q_gas	I₂+	I₂+	I₂+	I₂+	I₂+	I₂	I₂
Q_ch4	I₂+	I₂+	I₂+	I₂+	I₂	I₁	I₁
Online VFA	I₁	I₁	I₁	I₁	I₁	I₁	I₂+
Hopper load cell								D
Pre-pit level / pump counter									D

Charge balance + gas phase¶

Sensor	S_cation	S_anion	S_va_ion	S_bu_ion	S_pro_ion	S_ac_ion	S_hco3	S_nh3	p_gas_h2	p_gas_ch4	p_gas_co2	pTOTAL
Q_gas							I₁		I₁	I₁	I₁	D
Q_ch4							I₂		D	D	I₁	D
Q_co2							I₁		D	I₁	D	D
H₂ gas									D			I₁
p_head												D
pH (online)	D	D	D	D	D	D	D	D
Conductivity	C	C
Online VFA

Lab analytics¶

Dissolved components (0-11)¶

Lab quantity	S_su	S_aa	S_fa	S_va	S_bu	S_pro	S_ac	S_h2	S_ch4	S_co2	S_nh4	S_I
VFA sum (FOS)				D-Σ	D-Σ	D-Σ	D-Σ
TAC				D	D	D	D			D	D
FOS/TAC				D	D	D	D			D	D
pH (lab)											D
NH₄-N (lab)											D
Individual VFA (HPLC)				D	D	D	D
TKN											D-Σ
TS / VS (lab)												I₂
COD total	I₁(Σ)	I₁(Σ)	I₁(Σ)	I₁(Σ)	I₁(Σ)	I₁(Σ)	I₁(Σ)	I₁(Σ)	I₁(Σ)			I₁(Σ)
COD dissolved	D-Σ-diss	D-Σ-diss	D-Σ-diss	D-Σ-diss	D-Σ-diss	D-Σ-diss	D-Σ-diss	D-Σ-diss	D-Σ-diss			D-Σ-diss
Anion inventory
Cation inventory
Substrate TS/oTS/COD	(s_in[*] — influent, not state directly)

Biomass + charge balance (lab)¶

Lab quantity	X_PS_ / X_PF_ / X_S_*	X_I	X_su-h2	S_cation	S_anion	S_*_ion	S_hco3	S_nh3
TS / VS	I₁(Σ)	D	I₂+
COD particulate	D-Σ-part.	D-Σ-part.	D-Σ-part.
TAC				D	D	D	D	D
TKN								I₁
Anion inventory					D	D?	D
Cation inventory				D				I₁

Note on anion/cation inventory: \(S_{cation}\) and \(S_{anion}\) are lumped in the model as kmol/m³. A species-resolved inventory (Na⁺, K⁺, …) gives the pool sum directly; individual species do not exist in the state vector.

Implications for the UKF¶

Rules of thumb¶

Only standard SCADA (Q_gas, CHP, T, hopper, pre-pit)
- directly estimable: pTOTAL, \(Q_{solid}, Q_{liquid}\) (augmented)
- 1-step: \(S_{h2/ch4/co2}, S_{hco3}, p_{gas,*}\)
- 2-step: \(X_{ac}, X_{h2}, S_{ac}\)
- Effectively 6-8 separable dimensions, \(X_{ac}\)/\(X_{h2}\) partially
  degenerate.
+ NDIR gas analytics (CH₄/CO₂ fraction online)
- \(X_{ac}\) vs \(X_{h2}\) separable (via CH₄/CO₂ ratio).
- \(S_{co2}\) vs \(S_{hco3}\) cleaner separation.
- Optional H₂ sensor: makes \(p_{gas,h2}\) directly observable.
+ pH probe
- 9 charge-balance states directly observable.
- Newton iteration in the update disappears — algebraically clean.
+ FOS/TAC routine lab (daily/weekly)
- VFA sum + alkalinity sum directly.
- Acetate-methanogen chain robustly observable.
- Early-warning indicators quantitatively available.
+ Individual VFA (HPLC, monthly)
- \(S_{ac}, S_{pro}, S_{bu}, S_{va}\) individually rather than as sum.
- Separates acetoclast vs propionate-driven inhibitions.
+ NH4-N + TKN
- \(S_{nh4}\) directly, NH3 inhibition \(I_{nh3}\) quantifiable.
- Decisive for nitrogen-rich substrates (poultry manure, abattoir
  waste).
+ Cation/anion inventory (monthly)
- \(S_{cation}, S_{anion}\) directly — otherwise very hard to observe.

What the UKF cannot do, regardless of instrumentation¶

Separate individual ch/pr/li fractions. Disintegration and
hydrolysis rates act on sums; the split remains prior-determined — solution only via qPCR or substrate profiles.
\(X_I\)/\(S_I\) accumulation in hours. Inert pools have time constants
of weeks to months — no innovation, no update.
Substrate-specific disintegration online. Even with lab VFA and
pH, the split between X_PS and X_PF (slow/fast) remains weakly observable. Solution: the calibration artifact sets these parameters, the filter does not co-estimate them.
Individual biomass populations without qPCR — only summary
effects through the gas-flow response are measured.

Latency assessment¶

Measurement set	Update rate	Reaction time to plant upset
Q_gas only	seconds	minutes — very fast
+ NDIR + pH	seconds	immediate on pH shift
+ FOS/TAC routine lab	daily	1-2 day delay — UKF must bridge with prior
+ Individual VFA / NH4-N / cations	weekly-monthly	very slow correction; more a bias anchor than reaction
+ qPCR	annually or rare	validation, no reaction

STRIKE-GOLDD applicability for ADM1da¶

Short: not directly for the full 41-state model.

Reference data from the literature¶

Hellmann et al. (2023) empirically show, using the STRIKE_GOLDD toolbox (Matlab):

Model class	States	FISPO time	ORC-DF time
ADM1-R4	11	3 s	7 s
BMR3+ABC	13	12 s	5 s
ADM1-R3	17	11,959 s (≈ 3.3 h)	811 s
ADM1-R2	>17	no result (toolbox aborts)	no result

Our own Python/sympy benchmark¶

We rebuilt the STRIKE-GOLDD algorithm directly in Python/sympy (observability_experiment/subsystem_checker.py): symbolic Lie derivatives + numerical rank check at a random sample point (Sedoglavic 2002). Exactly what ORC-DF does internally, but without the Matlab↔Octave↔Python bridge.

A direct analysis of all 41 states is not practical: Lie-derivative complexity grows exponentially (a polynomial fit from synthetic test runs with n = 4, 6, 8 gives \(t \propto 4.04^{n}\) seconds and \(m \propto 3.39^{n}\) MiB; already at n = 16 the RAM demand exceeds 256 GiB). Hellmann (2023) stops at n = 17 for the same reason.

Solution: divide-and-conquer over five subsystems + one super-subsystem. The full model is decomposed along its topological structure (gas / acidogenesis / hydrolysis / charge balance / nitrogen) into blocks; each block is analysed in isolation, couplings are resolved via known inputs.

Results (Phase-1 sensor set: \(Q_{gas}\) + CH4/CO2 NDIR + pH online + FOS/TAC)¶

Subsystem	n	n_out	iters	rank	verdict	wall
A — Gas + methanogenesis	11	3	3	11	observable	0.7 s
B — Acidogenesis	9	1	5	6	partial (sympy toolchain limit; ceiling ≈ 9)	51 min
C — Disintegration / hydrolysis	10	3	12	7	partial (PS/PF split truly non-separable)	1.4 s
D — Charge balance / pH	8	2	3	8	observable	5.6 min
E — Nitrogen + \(S_I\)	2	2	0	2	observable (open-loop)	< 0.1 s
A+D fused (Variante II)	18	5	3	18	observable, no handshake assumptions	17.8 min

Total: ≈ 80 min compute, all runs inside the 1-hour-per-subsystem budget.

Key findings:

A+D fused is the strongest composite proof: 18 of 41 states,
including the full pH algebra \(S_H = (-fixed + \sqrt{fixed^2 + 4 K_w})/2\) and the inhibition factors \(I_{ac}, I_{h2}, I_{HAc}, I_{nh3}\), are structurally observable without any "opaque" boundary inputs. So the gas phase, methanogenesis, and the complete acid-base chemistry are provably estimatable.
C exposes the real structural deficit: TS and VS alone cannot
separate carbohydrates, proteins and lipids (rank 4/10). With COD-weighted measurement (1.03 / 1.5 / 2.9 gCOD/gVS for ch/pr/li) the rank rises to 7/10; the remaining three deficits are the PS/PF splits (slow vs. fast degradable), which are fundamentally not resolvable from process measurements and must be fixed by substrate characterization in the lab.
B is toolchain-limited, not physically limited: \(S_{fa}, X_{fa}\)
were removed as analytically decoupled (they flow via \(\rho_{fa}\) into \(S_{ac}, S_{h2}\) and thus into A, not into the VFA sum that FOS measures). The rank growth \(1 \to 2 \to 3 \to 4 \to 5 \to 6\) is steady at +1 per iteration; at iter 6 SymPy hits a CPython buffer limit (bytesobject.c:3219). The pattern strongly suggests the structural rank is 9, but only rank 6 is provable inside the current toolchain.
E (S_nh4, S_I) is open-loop observable: every RHS element is
available via A+B+C+D, so the output Jacobian already reaches rank 2 without Lie iteration. But without an NH4-N measurement there is no innovation channel; drift in the UKF must be caught by an OU prior.

Composite balance:

Block	Provably observable	Plausibly observable
A+D fused	18 / 18	18 / 18
B (substrate side)	5 / 9	9 / 9 (sympy limit)
C	7 / 10	7 / 10 (PS/PF deficit is real)
E	2 / 2 (open-loop)	2 / 2
ADM1da total	32 / 41	36 / 41

The remaining 5–9 states are: three PS/PF splits in C (lab characterization required), two nitrogen states in E (correction channel only with NH4-N), and 0–4 biomass states in B (structurally likely observable, unprovable inside the current toolchain).

Full details and reproduction instructions: observability_experiment/results.md.

Consequence for UKF design¶

Direct innovation channel for 30 states: 18 A+D + 5
acidogenesis substrates + 7 hydrolysis modes.
Open-loop propagation for 11 states: 4 acidogenesis biomasses
(X_su, X_aa, X_c4, X_pro), 3 PS/PF splits in C, 2 nitrogen states in E, 2 FA states (S_fa, X_fa observed slowly via A). Model them as OU-drift channels in StateVectorSpec.

Sources¶

pyadm1/core/adm1.py (v0.3.4) — all line citations refer to this.
pyadm1/components/biological/digester.py — VFA/TAC computation.
Hellmann, S. et al. (2023). Observability and Identifiability
Analyses of Process Models for Agricultural Anaerobic Digestion Plants. arXiv:2301.05068v3.
Haugen, F. et al. (2014). State Estimation and Model-Based Control of
a Pilot Anaerobic Digestion Reactor. J. Control Sci. Eng., 572621.
Villaverde, A. F. (2022). STRIKE_GOLDD 4.0. arXiv:2207.07346.
Wolf, C., Gaida, D., Bongards, M. (2014). Online-measurement systems
for agricultural and industrial AD plants — a review and practice test. Compendium :metabolon.
Step 1: Observability literature review

Sensor–state dependencies in ADM1da¶

Methodology¶

Notation and symbols¶

State classes¶

Subscripts (compound classes)¶

Observable quantities (outputs / measurement models)¶

Reaction rates \(\rho\) (in the code: Rho_*)¶

Inhibition factors \(I\) (values in [0, 1])¶

Key parameters¶

Measurement-model equations (from the code)¶

Gas volume flows¶

pH¶

VFA and TAC¶

A — Online measurements¶

A.1 Standard SCADA (most plants)¶

A.2 Online gas analytics (NDIR multi-gas sensor)¶

A.3 Online liquid-phase (premium instrumentation)¶

B — Lab analytics¶

B.1 Standard AD routine (typ. weekly)¶

B.2 Extended analytics (typ. weekly to monthly)¶

B.3 Substrate analytics (for s_in of the influent balance)¶

C — Research / specialized methods (for completeness)¶

ODE coupling structure — who depends on whom¶

Gas-phase ODEs (L. 914-918)¶

Methane chain (S_ch4 ODE, L. 828-834)¶

Acetate chain (S_ac ODE, L. 805-815)¶

Nitrogen chain (S_nh4 ODE, L. 840-853)¶

Particulate pools (L. 856-893)¶

Master table: sensor → exposed states¶

Online sensors¶

Dissolved components (0-11)¶

Biomass + augmented inputs¶

Charge balance + gas phase¶

Lab analytics¶

Dissolved components (0-11)¶

Biomass + charge balance (lab)¶

Implications for the UKF¶

Rules of thumb¶

What the UKF cannot do, regardless of instrumentation¶

Latency assessment¶

STRIKE-GOLDD applicability for ADM1da¶

Reference data from the literature¶

Our own Python/sympy benchmark¶

Results (Phase-1 sensor set: \(Q_{gas}\) + CH4/CO2 NDIR + pH online + FOS/TAC)¶

Consequence for UKF design¶

Sources¶

Reaction rates \(\rho\) (in the code: `Rho_*`)¶