The batch (bioreactor) simulator#

A “golden batch” is the run where everything went right, and batch automation commonly responds by replaying its charging recipe and setpoint schedules for every future batch. Replaying is open-loop control: it assumes the disturbances present during the golden run will recur, and they do not. The process_improve.simulation.batch module exists to make the consequences of that assumption measurable, with numbers that reproduce exactly from a seed:

  1. Replaying a golden schedule does not reproduce the golden outcome.

  2. The spread persists even when the measured initial conditions are held identical, because disturbances also arise while the batch runs.

  3. The quality variance under replay therefore splits into a share that adapting the schedule before the batch could address, a share observable only during the batch (the case for mid-course correction), and a noise floor.

Because the simulator’s own kinetics are known, the true optimal schedule for any batch’s initial conditions is computable. That gives later, data-driven schemes (latent-variable models, trajectory optimisation, mid-course correction) a ceiling to be scored against, which no historical dataset can offer.

The process model#

A 10-day fed-batch bioreactor, sampled twice daily: biomass, substrate, product (titer) and volume, with pH and temperature as the manipulated schedule and dissolved oxygen, offgas CO2 and volume as recorded responses. Growth follows the gamma-concept cardinal temperature and pH model of Rosso et al. (1995); production follows Luedeking and Piret (1959). Three couplings give the schedule a real, condition-dependent optimum: an oxygen-transfer ceiling with hypoxic death, strong hypothermic growth arrest (the reason the industrial biphasic temperature shift works), and substrate consumption by production with starvation death. The full equations, parameter meanings and citations are in the module docstring (process_improve.simulation.batch).

Three disturbance channels can each be scaled or switched off independently: measured initial conditions (an 11-variable upstream Z block with three feed classes A, B and C), an unmeasured within-batch disturbance that is visible in the gas trajectories, and control-loop plus measurement noise at instrument scale. All random draws are made on every call and multiplied by their channel scale, so the same seed with one channel switched off is a true counterfactual for the same batch.

Realism: the sensitivity budget#

A simulator whose quality output moves when an input moves by less than an instrument can resolve is not believable. The nominal schedule therefore sits at stationary points of the response (the pH optimum and the interior optimum of the production-phase temperature hold), so instrument-scale deviations are second order, while sustained multi-degree deviations cross real mechanisms and cost real titer. Measured on the default configuration (nominal titer 8.01 g/L, 200 noise replicates, seed 0):

Perturbation

Effect on titer

Control-loop noise, sd 0.15 degC and 0.02 pH

sd 0.25%

Sustained bias of +0.1 / -0.1 degC

-0.22% / -0.20%

Sustained bias of +0.02 / -0.02 pH

-0.01%

One 12-hour sample at +0.5 degC

-0.30%

Sustained bias of +1.0 / -1.0 degC

-11.5% / -7.1%

Sustained bias of +2.0 / -2.0 degC

-20.6% / -22.5%

These numbers are enforced as acceptance tests in tests/test_simulation_batch.py, and sensitivity_budget() recomputes the table from the live configuration, so the claim can be checked against any parameter changes rather than taken on trust:

from process_improve.simulation import BioreactorSimulator

sim = BioreactorSimulator()
print(sim.sensitivity_budget(random_state=0))

The three claims, in code#

Replaying the schedule does not reproduce the outcome:

campaign = sim.simulate_campaign(50, policy="replay", random_state=0)
titer = campaign.quality["titer"]
print(titer.mean(), titer.std(ddof=1))   # a spread of roughly 13% CV

Holding the measured initial conditions identical does not remove the spread. dataclasses.replace derives a configuration with the initial-condition channel off; what remains (close to 7% CV, about thirty times the noise floor) arose during the batches:

import dataclasses
from process_improve.simulation import BioreactorConfig

same_z = BioreactorSimulator(
    dataclasses.replace(BioreactorConfig(), ic_scale=0.0, noise_scale=0.0)
)
same_z_campaign = same_z.simulate_campaign(50, policy="replay", random_state=0)

And the decomposition, which runs the four campaigns and reports the interaction residual of the nonlinear model explicitly instead of forcing the buckets to sum:

from process_improve.simulation import variance_decomposition

print(variance_decomposition(sim, n_batches=200, random_state=0))

The golden batch and the adaptation ceiling#

golden_trajectory() finds the true optimal schedule for the nominal initial conditions by optimising over the simulator’s own kinetics; it recovers the industrial biphasic shape (a warm growth phase near 37.5 degC, a smooth downshift into a production hold near 29 degC) without being told it. On the defaults it reaches 8.54 g/L against 8.01 g/L for the built-in nominal recipe.

optimal_trajectory() does the same for any batch’s own initial conditions. The gap between replaying the golden schedule and each batch’s own optimum is the value a perfect feedforward adaptation could recover; on the default configuration it ranges from about 3% at the centre of the good feed class to over 40% at the centre of the poorest. The "adapted" campaign policy runs every batch at its own optimum, as that ceiling, and the "historical" policy adds deliberate setpoint variation, since a perfectly consistent history carries no information about how the controls affect quality.

golden = sim.golden_trajectory()
adapted = sim.optimal_trajectory(campaign.initial_conditions.iloc[0])

Campaign outputs use the package’s standard batch dictionary format (dict[batch_id, DataFrame]), so they feed directly into process_improve.batch.data_input.dict_to_wide() and the alignment tooling for latent-variable modelling.

Agent tools#

Two registered tools expose the baseline to agent callers: simulate_batch_campaign (campaign outcomes with the disturbance-free reference titer for comparison) and decompose_batch_quality_variance (the variance split). The golden_batch_baseline analysis recipe chains them into the three-step argument above.

Note

The accompanying book, Process Improvement using Data, develops the golden-batch discussion and the latent-variable methods this baseline is built to support.