Analytical guide shows when rooftop solar and batteries should act together or separately under time‑of‑use net‑energy‑metering tariffs
arXiv: 2610.40650v1
The research presents an exact analytic solution for a prosumer’s joint decision to install rooftop photovoltaic (PV) panels and battery storage under a time‑of‑use net‑energy‑metering (ToU‑NEM) tariff. The solution divides tariff parameters into six distinct regions, each revealing whether PV and storage complement or substitute one another. In regions where the off‑peak supply surplus is high relative to peak prices, storage raises PV value; in other regions, the opposite occurs.
Table of optimal PV capacities by tariff region, displaying which PV and storage combinations are optimal under each region.
Reg.
Cond. on cg
g∗
Reg.
Cond. on cg
g∗
(A)
cg>Φ++
0
(D)
cg>Φ++
0
Φ++>cg>Φ−+
goff(0)
Φ++>cg>Φ0+
goff(0)
cg<Φ−+
g¯
Φ0+>cg>Φ0−
gnz
Φ0−>cg>Φ−−
goff(z¯)
cg<Φ−−
g¯
(B)
cg>Φ++
0
(E)
cg>Φ++
0
Φ++>cg>Φ0+
goff(0)
Φ++>cg>Φ+0
[gon(z¯)]+
Φ0+>cg>Φ−0
gnz
Φ+0>cg>Φ0−
gnz
cg<Φ−0
g¯
Φ0−>cg>Φ−−
goff(z¯)
cg<Φ−−
g¯
(C)
cg>Φ++
0
(F)
cg>Φ++
0
Φ++>cg>Φ+0
[gon(z¯)]+
Φ++>cg>Φ+−
[gon(z¯)]+
Φ+0>cg>Φ−0
gnz
Φ+−>cg>Φ−−
goff(z¯)
cg<Φ−0
g¯
cg<Φ−−
g¯
TABLE I. TABLE I shows the optimal PV capacities and storage decisions for each of the six tariff‑dependent regions, illustrating when PV and storage act as complements or substitutes.Lado, Nathan Engelman, Alahmed, Ahmed, Botterud, Audun, Amin, Saurabh · arXiv:2610.40650v1 · TABLE I: · CC BY 4.0
The problem is framed as a cost‑minimization program that includes charging/discharging schedules, PV size, and storage size. By exploiting the ordering of import and export tariffs, the bilinear terms are linearized, yielding a tractable linear optimization that gives closed‑form expressions for the optimal operating and investment choices.
A simulation using 500 Massachusetts single‑family home profiles validates the analytical findings. Hourly energy data from 2018 were aggregated into peak and off‑peak periods, and PV and battery costs were set at 2025‑dollar benchmark levels. Three tariff designs—P, O, and B—were tested. The results confirm the predicted complementary and substitutive interactions, and illustrate how varying PV or battery costs changes adoption patterns and the utility’s net‑cost outcome.
Fig. 6. Fig. 6 illustrates how varying PV and storage costs affect optimal PV investment, storage adoption, and distribution‑utility net‑cost change for the three tariff scenarios in the Massachusetts case study.Lado, Nathan Engelman, Alahmed, Ahmed, Botterud, Audun, Amin, Saurabh · arXiv:2610.40650v1 · Fig. 6: · CC BY 4.0
The analysis assumes perfect forecasting of demand and solar generation, no network constraints, and that battery power capacity is unconstrained. These simplifications mean the conclusions may not hold under real‑world operating uncertainty.
**Figure 6** is used to show how changes in PV and storage costs affect optimal adoption and the distribution utility’s net‑cost change across the three tariff scenarios, directly supporting the validation paragraph above.
Analytical guide shows when rooftop solar and batteries should act together or separately under time‑of‑use net‑energy‑metering tariffs | arXiv News
