Industry: Utilities Product: ModelRisk Application: Renewable energy adoption
A distribution utility serving 1.35 million eligible owner-occupied premises had a single S-curve in its integrated resource plan: rooftop-solar and behind-the-meter DER would reach about 34% penetration by year 10, and the low-voltage feeders would need reinforcement somewhere "in the early 2030s." That single curve drove a capital programme worth hundreds of millions of dollars. The problem was that the curve had no width. Adoption of a network technology is driven by word-of-mouth imitation, by an uncertain ultimate market size, and by a net-metering policy regime that can accelerate or stall the whole trajectory -- none of which the planners could pin to a point.
The team rebuilt the forecast in ModelRisk as a Bass diffusion model with uncertain innovation, imitation and market-potential parameters, plus a persistent policy regime that shifts the entire curve. The first output was not a date -- it was the fan of adoption paths below, and the distribution of the year in which penetration crosses the level at which feeders start to need reinforcement.
The deterministic plan put year-10 penetration at 34.2%. The simulation agreed on the centre -- simulated P50 = 33.2% -- but revealed a band the single curve hid: P10 = 19.2% to P90 = 50.9% at year 10. Reinforcement that the plan scheduled for one date could arrive years early or years late, and the capital sequencing has to be robust across that whole spread.
A Bass curve has three parameters, and a deterministic forecast freezes all three. The innovation coefficient p sets how fast early adopters move; the imitation coefficient q sets how strongly each new installation pulls in neighbours; the ultimate potential m sets how high the curve ever goes. Pick central values and you get one tidy S. But q is the dominant term and the least knowable, m depends on how many roofs are ultimately suitable and financeable, and a change in net-metering buyback rules shifts q for the rest of the trajectory -- not for one year. The model treats p as LogNormal, q as LogNormal multiplied by a persistent policy regime drawn once per path, and m as a Beta fraction of eligible roofs. Because the regime is persistent, multi-year penetration keeps its spread instead of averaging back to the mean -- the same reason a wind forecast does not collapse to its long-run average.
Feeders begin to need reinforcement -- reverse power flow and voltage rise on the LV network -- once local DER penetration passes roughly 25%. Plotting the year-10 penetration distribution against that line turns a planning assumption into a probability.
With a median of 33.2% and a P90 of 50.9%, the distribution sits mostly above the threshold: there is a 75.6% probability that year-10 penetration already exceeds the 25% reinforcement trigger. The deterministic 34.2% looked comfortably above the line too -- but it could not say how often the line is crossed, and it is the probability, not the point, that sizes the contingency in the capital plan.
The capital question is timing: by which year is reinforcement more likely than not? Sweeping the cumulative probability that penetration has crossed 25% by each year answers it directly.
The crossing probability reaches 50% at year 9, and by the end of the 15-year horizon 99.2% of paths have crossed -- reinforcement is effectively certain within the plan, the only open question is when. Among the paths that cross, the median crossing year is 9 and the P10 is year 7: in one path in ten, the feeders need reinforcing by year 7, two to three years ahead of the deterministic schedule. That early tail is what justifies pre-ordering long-lead transformers rather than waiting for the central-case date.
If the band on year-10 penetration is what creates planning risk, the obvious follow-up is which input drives that band. The tornado ranks the one-at-a-time P10-to-P90 swing of each input on year-10 penetration.
The imitation coefficient q moves year-10 penetration by 26.8 percentage points across its range -- by far the largest driver -- followed by the persistent policy regime at 19.7 points and ultimate market potential at 14.0 points. The innovation coefficient p matters least (12.4 points). This re-ranking is the actionable finding: the single most valuable thing the utility can do to narrow the forecast is to reduce uncertainty about word-of-mouth dynamics and to stabilise the net-metering policy regime -- not to refine the early-adopter rate the deterministic model had agonised over.
For a network technology, "adoption" is not a curve, it is a fan -- and the decision that matters, when to reinforce the feeders, lives in the distribution of the crossing year, not in the date where a single S-curve happens to cross the line.