Stacked Revenue Model Overview

• Stacked revenue model is a compositional approach that explicitly decomposes revenue into separate components, such as retention and monetization, to enhance forecasting accuracy.
• In cohort analysis, it couples Bayesian retention submodels with linear revenue models to estimate expected revenue via customer count, retention probabilities, and per-user income.
• In grid-connected battery systems, it aggregates revenues from energy arbitrage, rental services, and ancillary streams, enabling joint optimization in market and operational settings.

A stacked revenue model is a composite formulation in which revenue is represented by explicitly combining multiple component processes rather than by a single undifferentiated equation. In the cohort-analysis setting, it couples a retention submodel and a revenue submodel so that expected cohort revenue is driven jointly by active-user counts and per-user monetization (Orduz, 22 Apr 2025). In grid-connected battery systems, the same term denotes the aggregation of multiple storage and flexibility value streams, such as day-ahead energy arbitrage and Battery-as-a-Service rental income, within a single optimization objective (Pocola et al., 6 Jul 2025). This suggests that the expression is application-dependent, but in both uses it refers to structured composition: separate mechanisms are modeled explicitly and then combined to obtain revenue, forecasts, or control decisions.

1. Conceptual scope and domain-dependent meaning

In the cohort-revenue literature, the stacked construction is probabilistic and hierarchical. The retention component models the probability that a customer in cohort $c$ remains active at time $t$, while the revenue component models the $(\log)$ revenue per active user at the same cohort-time index. The two components are then combined multiplicatively through cohort size, retention probability, and conditional per-user revenue to obtain expected total revenue (Orduz, 22 Apr 2025).

In the battery-systems literature, the stacked construction is operational and market-facing. A grid-connected battery operator may derive value from energy arbitrage in the day-ahead market, Battery-as-a-Service rental income from an energy community, and, optionally, ancillary services or peak-shaving revenues. The stacked revenue model therefore refers to aggregation of multiple revenue or cost-saving streams into one annual objective (Pocola et al., 6 Jul 2025).

A common source of ambiguity is the assumption that “stacked revenue model” has a single canonical meaning. The literature instead uses the term in at least two distinct but structurally related senses. One sense is model stacking across statistical subproblems; the other is revenue stacking across market or service streams. The unifying feature is explicit decomposition followed by recombination.

2. Bayesian cohort-revenue formulation

The cohort formulation indexes cohorts by $c$ and time-periods by $t$. Cohort $c$ has size $N_n^c$, interpreted as customers acquired at $t=0$, and $N^a_{c,t}$ denotes the observed number of active customers in period $t$. The latent retention probability is

$t$0

Retention is modeled with a Bayesian additive regression trees ensemble,

$t$1

linked to probability through the logistic map,

$t$2

For each tree $t$3, the structure prior assigns a split probability $t$4 at depth $t$5, so that

$t$6

and leaf parameters satisfy

$t$7

Conditional on $t$8, the observed active counts follow

$t$9

with retention likelihood

$(\log)$0

The revenue component is linear-Gaussian. Let $(\log)$1 be the $(\log)$2 revenue per active user in cohort $(\log)$3 at time $(\log)$4, and let $(\log)$5 denote covariates such as standardized “cohort-age,” “age,” “month,” and interactions. Then

$(\log)$6

with priors

$(\log)$7

Its likelihood is

$(\log)$8

The stacking mechanism combines the two components through

$(\log)$9

In this formulation, revenue is not estimated directly as a single response. It is obtained by coupling survival-like retention behavior with conditional monetization, which preserves separate structure for each process (Orduz, 22 Apr 2025).

3. Joint inference, uncertainty quantification, and interpretability

The cohort model is implemented as a single joint Bayesian program, for example in PyMC or NumPyro, and all unknowns can be sampled together via MCMC using NUTS or approximated via stochastic variational inference. The joint posterior is

$c$0

Under posterior sampling, one draws

$c$1

and computes, for each draw,

$c$2

The empirical distribution of $c$3 yields the posterior predictive distribution for $c$4, and highest-density intervals or quantile-based intervals are read off from these draws. The paper emphasizes that the framework can forecast future revenue and retention rates with well-calibrated uncertainty through highest density intervals and can enable coherent out-of-sample forecasts with credible intervals (Orduz, 22 Apr 2025).

Forecasting to $c$5 uses the fact that $c$6 and $c$7 are functions of known “cohort-age” and calendar “month.” Future $c$8 and $c$9 are plugged in to obtain draws of $t$0 and $t$1, after which $t$2 draws are formed in the same way. Uncertainty naturally grows as the input moves further out-of-sample.

Interpretability is distributed across the two submodels. BART-based partial dependence plots and individual conditional expectation plots show how $t$3 varies as one feature changes while others are held fixed. ICE lines for each $t$4 together with the average PDP curve help detect non-linearities and interactions. Variable-importance in BART can be computed as the percentage contribution of each covariate to the in-sample $t$5 of the retention submodel. For the linear revenue component, standard trace plots, posterior-predictive checks such as overlays of observed $t$6 versus simulated $t$7, and posterior intervals on $t$8 are used to judge fit.

The framework is also explicitly extensible. Additional covariates, including marketing-channel and macro-indicators, can be appended to $t$9 in the BART retention model and to $c$0 in the linear revenue model. BART will non-parametrically discover interactions, whereas the linear part remains interpretable (Orduz, 22 Apr 2025).

4. Grid-connected batteries and stacked service revenues

In battery energy storage systems, a stacked revenue model aggregates distinct storage and flexibility services into a unified annual objective. The formulation described for a grid-connected battery operator with part of its capacity rented to an energy community includes energy arbitrage in the day-ahead market, Battery-as-a-Service rental income from the energy community, and, optionally, ancillary services or peak-shaving revenues (Pocola et al., 6 Jul 2025).

Using $c$1 for net discharge at time $c$2 in the day-ahead market and $c$3 for effective service usage assigned to the community at time $c$4, total annual revenue can be written as

$c$5

or, if per-timestep community savings at price $c$6 are included,

$c$7

The associated battery-control problem is formulated as a linear program. For each timestep $c$8, decision variables are charging power $c$9, discharging power $N_n^c$0, state of charge $N_n^c$1, imported energy $N_n^c$2, and exported energy $N_n^c$3. Parameters include time-step length $N_n^c$4, efficiencies $N_n^c$5, usable capacity $N_n^c$6, maximum charge or discharge rate $N_n^c$7, bounds $N_n^c$8 and $N_n^c$9, and prices $t=0$0 and $t=0$1.

The optimization objective is

$t=0$2

The constraints enforce state-of-charge evolution,

$t=0$3

grid-interface balance,

$t=0$4

capacity limits $t=0$5, power limits $t=0$6, an optional end-of-day constraint $t=0$7, and an optional cycle-wear constraint,

$t=0$8

Here, “stacked” refers to combining market uses and contractual service provision in a single operator objective, rather than to statistical coupling as in the cohort formulation.

5. Regularization, BaaS pricing, and case-study results

The battery paper emphasizes that existing approaches for battery control with daily time windows have a number of important limitations in practical deployments and introduces regularization functions to address them (Pocola et al., 6 Jul 2025). Two penalty terms are added to the objective:

$t=0$9

which discourages unnecessary charge/discharge activity, and

$N^a_{c,t}$0

which encourages carrying some energy into the next day. Typical values reported from the case study are $N^a_{c,t}$1 and $N^a_{c,t}$2.

Sizing and pricing in the Battery-as-a-Service model are expressed through community cost savings and operator opportunity cost. With rented capacity $N^a_{c,t}$3 and rental price $N^a_{c,t}$4, community savings are

$N^a_{c,t}$5

so the community will pay any price satisfying $N^a_{c,t}$6. The operator’s opportunity cost is

$N^a_{c,t}$7

so the operator requires $N^a_{c,t}$8. The feasible mutually beneficial region is therefore

$N^a_{c,t}$9

and the equilibrium price $t$0 can be negotiated anywhere in that interval.

The case study uses a community of 200 households with 840 MWh/yr demand and a 330 kW Enercon E-33 wind turbine, with $t$1. Tariff settings include flat imports $t$2 with exports $t$3 or $t$4, and dynamic imports $t$5 with exports $t$6 capped at $t$7. Market data are day-ahead prices for the Netherlands in 2023 from ENTSO-E.

Under flat import $t$8 and no export, community savings peak at $t$9, annual net savings $t$00 are approximately $t$01, and the operator’s day-ahead revenue loss $t$02 is approximately $t$03, yielding a feasible rental price between about $t$04 and $t$05. The abstract states that a community of 200 houses with a 330 kW wind turbine can save up to 12,874 euros per year by renting just 280 kWh of battery capacity after subtracting battery rental costs. Under dynamic pricing with export, the $t$06-regularized LP control outperforms all benchmarks, achieving both lower costs for the community and reduced battery wear compared to unregularized or heuristic methods.

6. Comparative structure, significance, and interpretive cautions

The two formulations share a structural principle but differ in mathematical composition. In the cohort model, the stacking mechanism is multiplicative:

$t$07

In the battery model, the stacked objective is additive across service streams and timesteps:

$t$08

This suggests that “stacked” does not prescribe a single algebraic template. Instead, it indicates that revenue-relevant substructures are modeled separately and then combined according to the application’s physics, market design, or customer process.

The inferential and operational roles also differ. In the Bayesian cohort setting, the principal gains are uncertainty quantification, partial dependence and individual conditional expectation analysis, and coherent forecasting with highest-density intervals or quantile-based intervals. The paper further states that Bayesian inference unifies the components, automatically sharing statistical strength across cohorts and quantifying uncertainty end-to-end (Orduz, 22 Apr 2025). In the battery setting, the principal gains are optimization over multiple value streams, explicit accounting for opportunity cost and mutually beneficial pricing, and operational safeguards against excessive cycling and end-of-day emptying (Pocola et al., 6 Jul 2025).

A common misconception is that stacked revenue modeling is synonymous with adding more revenue lines to a spreadsheet objective. The cohort formulation shows that stacking can instead mean coupling heterogeneous probabilistic submodels inside a joint posterior program. Conversely, the battery formulation shows that stacking can refer to economically combining distinct storage and flexibility services even when the control problem remains linear. The literature therefore supports a broader interpretation: stacked revenue modeling is a compositional methodology whose concrete form depends on whether the primary problem is statistical estimation, forecasting, market optimization, or asset control.