Methodology | Voltary

Overview

Voltary simulates household load, PV generation, battery dispatch, and tariff settlement at fixed time intervals. The results are intended for robust comparisons across scenarios and configurations, not as a guarantee that realized savings will match the model exactly.

On this page, we document the currently implemented model logic through formulas, assumptions, data sources, and model boundaries.

Notation and units

The notation below is used consistently across the page. Intermediate state markers such as (sd), (reserve), (d), and (pv) denote temporary SoC states inside a single dispatch interval.

Symbol	Unit	Meaning
$E$	kWh	Energy quantity
$P$	kW	Power
$p$	EUR/kWh	Price or tariff component
$C$	EUR	Cost or annual savings
$I$	EUR	Investment
$\Pi$	EUR	Total profit over the analysis horizon
$\eta$	1	Efficiency factor
$S$	kWh	Battery state of charge
$t$	index	Simulation interval
$Y$	years	Analysis horizon

Indices, suffixes, and arrow notation

Notation	Meaning	Example
$\mathrm{gross}$	energy before efficiency losses	$E^{\mathrm{grid}\rightarrow\mathrm{bat},\mathrm{gross}}$ is gross energy charged from the grid.
$\mathrm{net}$	energy after efficiency losses or expressed as a net effect	$E^{\mathrm{charge},\mathrm{net}}$ is energy stored net inside the battery.
$\mathrm{res}$	remaining quantity after an earlier step	$L^{\mathrm{res}}$ is residual load after direct PV coverage.
$\mathrm{exp,pre}$	potential export before regulatory limitation	$E^{\mathrm{exp,pre}}$ is only later capped to $E^{\mathrm{exp}}$ if required.
$\mathrm{dyn}\;/\;\phi_{\mathrm{fi}}$	dynamic tariff notation or feed-in scaling reference	$p^{\mathrm{dyn}}$ is the interval price; $\phi_{\mathrm{fi}}$ scales permitted feed-in.
$a \rightarrow b$	directed energy flow from source a to sink b	$\mathrm{pv}\rightarrow\mathrm{load}$ means direct PV-to-load coverage; $\mathrm{grid}\rightarrow\mathrm{bat}$ means battery charging from the grid.
$(\mathrm{sd}) / (\mathrm{reserve}) / (\mathrm{d}) / (\mathrm{pv})$	temporary battery state of charge within the same interval	$S_t^{(\mathrm{sd})}$ is after self-discharge, $S_t^{(\mathrm{reserve})}$ after reserve restoration, $S_t^{(\mathrm{d})}$ after discharge, and $S_t^{(\mathrm{pv})}$ after PV charging.

Simulation pipeline

The outputs are produced in a fixed sequence from data intake and normalization to physical simulation and economic evaluation.

1. Inputs

Consumption data, PV configuration, battery parameters, price assumptions, and tariff extras are assembled as model inputs.

2. Normalization

Time stamps, units, and missing values are aligned to a common interval structure so that load, PV, and price series can be compared directly. For uploads without an existing battery, we carry forward into later years the hourly pattern observed in the upload: how much PV directly covers load and in which hours residual load and grid import still remain despite PV. This matters because otherwise hours with strong PV but remaining residual load and grid import would be smoothed away. Those are exactly the spikes a battery can often cover later.

3. PV modelling

Depending on the workflow, we use uploaded PV data or compute weather-based PV production curves from PVGIS hourly data. We send the location, orientation, tilt, and allocated kWp per PV surface to PVGIS and use its documented default value of 14% system losses. For forecasts, we combine three real historical PVGIS years by taking the middle-yield month for each calendar month. In new-system scenarios where the selected PV size is smaller than the configured roof capacity, the highest-yield surfaces are allocated first; only those allocated partial surfaces feed into the PVGIS profile.

4. Battery dispatch

For each interval, SoC bounds, power constraints, efficiencies, and economic thresholds determine whether the battery charges, discharges, or remains idle. The battery therefore follows a transparent rule-based logic rather than a forecast-optimized controller.

5. Tariff settlement

Import cost, export revenue, and annual fixed components are aggregated under the active pricing model.

6. Evaluation

The model derives metrics such as self-sufficiency, self-consumption, ROI, payback, and profit and then ranks configurations by the chosen objective.

Battery model

The battery is modelled as a discrete-time storage problem with SoC bounds, charge/discharge power limits, and efficiency losses. Within each interval, the implementation follows this sequence: SoC clipping, direct PV-to-load allocation, self-discharge, reserve restoration, auxiliary demand, economically admissible discharge, PV charging, optional grid charging, and only then export or feed-in limitation.

(1)

Calendar aging

Usable capacity declines over time even without active cycling. The model captures this effect in a chemistry-aware way and makes it depend on the average SoC of the simulation day.

L_t^{\mathrm{cal}} = \sum_{\tau \le t} m_{\mathrm{cal}}\, a_{\mathrm{chem}}\, m_{\mathrm{soc}}\!\left(\bar s_\tau\right)\left(\sqrt{y_\tau} - \sqrt{y_{\tau-1}}\right)

Variables

L_t^{\mathrm{cal}}

cumulative calendar-driven capacity loss up to t

a_{\mathrm{chem}}, m_{\mathrm{soc}}(\bar s_t)

calendar-aging sensitivity multiplier together with chemistry-specific base coefficient and SoC multiplier

\bar s_t, y_t

average SoC of the simulation day and years elapsed up to t

Interpretation

In the model, calendar aging follows a square-root-of-time structure. Higher average SoC levels accelerate aging through the multiplier $m_{\mathrm{soc}}(\cdot)$ , while the base coefficients differ by cell chemistry. The user-facing multiplier $m_{\mathrm{cal}}$ scales this baseline term for sensitivity analysis without changing the underlying model structure.

The calendar-aging sensitivity is an optional high-level scenario control, not a manufacturer-specific cell coefficient.

(2)

Cycle aging

In addition to calendar aging, cycling reduces the available capacity. The daily SoC path is therefore translated into equivalent full cycles and a daily cycle depth.

n_t^{\mathrm{efc}} = \frac{1}{2}\sum_{i \in t}\left|s_i - s_{i-1}\right|

D_t = \max_{i \in t}\!\left(s_i\right) - \min_{i \in t}\!\left(s_i\right)

N_{\mathrm{life},t} = N_{\mathrm{cyc}}\, m_{\mathrm{depth}}(D_t)

L_t^{\mathrm{cyc}} = 0.20 \sum_{\tau \le t} m_{\mathrm{cyc}}\, \frac{n_\tau^{\mathrm{efc}}}{N_{\mathrm{life},\tau}}

Variables

n_t^{\mathrm{efc}}, D_t

equivalent full cycles and daily cycle depth of simulation day t

N_{\mathrm{life},t}, m_{\mathrm{depth}}(D_t)

chemistry- and depth-dependent effective cycle life

L_t^{\mathrm{cyc}}, N_{\mathrm{cyc}}

cumulative cycle-driven capacity loss up to t, guaranteed cycles, and the cycle-aging sensitivity multiplier

Interpretation

Shallow daily SoC swings consume less life in the model than deep cycles. The factor 0.20 maps cumulative cycle damage into capacity loss, so calendar and cycle components jointly determine the available capacity. The user-facing multiplier $m_{\mathrm{cyc}}$ scales the baseline cycle-damage term for sensitivity analysis.

Like the calendar-aging sensitivity, the cycle-aging sensitivity is an optional scenario setting rather than a raw internal coefficient table.

(3)

Available capacity and SoC bounds

The allowable operating band scales with the battery capacity still available at the relevant point in the simulation; minimum and maximum SoC are therefore not fixed kWh values.

E_{\mathrm{cap},t} = E_{\mathrm{cap},0}\left(1 - L_t^{\mathrm{cal}} - L_t^{\mathrm{cyc}}\right)

S_{\min,t} = \sigma_{\min} E_{\mathrm{cap},t}

S_{\max,t} = \sigma_{\max} E_{\mathrm{cap},t}

Variables

E_{\mathrm{cap},t}

available battery capacity in interval or simulation day t

L_t^{\mathrm{cal}}, L_t^{\mathrm{cyc}}

cumulative calendar- and cycle-driven capacity losses up to t

\sigma_{\min}, \sigma_{\max}

minimum and maximum allowed SoC shares, e.g. 10% and 95%

Interpretation

In the main implementation path, usable capacity is propagated over time with a chemistry-aware calendar and cycle aging model. The lower and upper SoC bounds are recomputed from that currently available capacity in each interval.

(4)

Per-interval power constraints

Charge and discharge power are first converted into the maximum amount of energy that can be moved during one interval. This limit applies in addition to the SoC bounds.

\bar E_t^c = P_c \Delta t

\bar E_t^d = P_d \Delta t

Variables

\bar E_t^c, \bar E_t^d

maximum chargeable and dischargeable energy in interval t

P_c, P_d

battery charge and discharge power

\Delta t

simulation interval length in hours

Interpretation

This makes short-term flexibility power-limited by construction. A large battery without sufficient power cannot react arbitrarily fast to price or load spikes.

(5)

Battery wear cost

The implementation spreads battery investment across nominal lifetime cycle throughput given by capacity times guaranteed cycles.

w_{\mathrm{bat}} = \frac{I_{\mathrm{bat}}}{E_{\mathrm{cap}} N_{\mathrm{cyc}}}

Variables

w_{\mathrm{bat}}

wear cost per internally cycled kWh

I_{\mathrm{bat}}

battery investment

E_{\mathrm{cap}}, N_{\mathrm{cyc}}

battery capacity and guaranteed cycles

Interpretation

This definition matches the current dispatch logic. Efficiency losses are not baked into wear cost; they are applied afterwards in the charge and discharge thresholds.

The reference quantity here is internally cycled kWh, not lifetime net energy delivered to the load. The current implementation deliberately uses a linear wear-cost model based on guaranteed cycles; additional aging drivers such as temperature, charge and discharge power, depth of discharge, and calendar aging are not modeled separately. A possible future extension could derive state-dependent marginal aging cost from the long-horizon chemistry-aware aging model, but the current dispatch path keeps wear cost as a simpler operational proxy.

(6)

Maximum charge price

Grid charging is only economical when the purchase price is low enough to cover both wear cost and round-trip losses.

p_{\mathrm{charge}}^{\max} = \frac{w_{\mathrm{bat}}}{\eta_c \eta_d}

Variables

p_{\mathrm{charge}}^{\max}

highest economical grid-charging price

w_{\mathrm{bat}}

battery wear cost

\eta_c \eta_d

round-trip efficiency

Interpretation

Under fixed pricing, this threshold is typically below the flat import price, so pure grid arbitrage remains disabled.

In the current implementation this threshold is derived from the operational wear-cost proxy above, not from a state-dependent chemistry-aware marginal aging model.

(7)

Minimum discharge price

Discharging only makes sense when the avoided import price exceeds wear cost after accounting for discharge losses.

p_{\mathrm{discharge}}^{\min} = \frac{w_{\mathrm{bat}}}{\eta_d}

Variables

p_{\mathrm{discharge}}^{\min}

lowest economical discharge price

w_{\mathrm{bat}}

battery wear cost

\eta_d

discharge efficiency

Interpretation

The battery therefore does not discharge mechanically at every opportunity. It only discharges in intervals where the modeled value of saved grid energy exceeds modeled wear.

As with the charge threshold, the current implementation uses the simpler operational wear-cost proxy here. A future extension could make this threshold depend on state-dependent marginal aging cost.

(8)

Direct PV coverage, residual load, and surplus

Before the battery reacts, household demand is first covered directly from concurrent PV generation. This yields the residual load for possible discharge and the PV surplus for later charging or export.

E_t^{\mathrm{pv}\rightarrow\mathrm{load}} = \min\!\left(E_t^{\mathrm{pv}}, E_t^{\mathrm{load}}\right)

L_t^{\mathrm{res}} = \max\!\left(E_t^{\mathrm{load}} - E_t^{\mathrm{pv}\rightarrow\mathrm{load}}, 0\right)

X_t = \max\!\left(E_t^{\mathrm{pv}} - E_t^{\mathrm{pv}\rightarrow\mathrm{load}}, 0\right)

Variables

E_t^{\mathrm{pv}}, E_t^{\mathrm{load}}

PV generation and household load in interval t

E_t^{\mathrm{pv}\rightarrow\mathrm{load}}

load covered directly by PV

L_t^{\mathrm{res}}, X_t

remaining residual load and remaining PV surplus

Interpretation

These definitions are the starting point of the dispatch sequence. All later battery decisions are based on residual load and surplus, not on gross load and PV values.

(9)

Self-discharge

Chemical idle losses are modeled separately from charge and discharge efficiency. They reduce stored energy in proportion to the current SoC even when the battery is otherwise inactive.

E_t^{\mathrm{self}} = \lambda_{\mathrm{sd}} \tilde S_{t-1}

S_t^{(\mathrm{sd})} = \tilde S_{t-1} - E_t^{\mathrm{self}}

Variables

E_t^{\mathrm{self}}

self-discharge loss in interval t

\lambda_{\mathrm{sd}}

compounded self-discharge factor derived from the monthly rate

S_t^{(\mathrm{sd})}

SoC immediately after applying self-discharge

Interpretation

This term captures chemically driven loss of stored energy rather than electrical conversion losses. It is applied before reserve restoration, auxiliary demand, or ordinary dispatch decisions.

(10)

Reserve restoration after self-discharge

If self-discharge pushes SoC below the configured minimum reserve, the model restores that reserve from PV surplus first and from grid second, subject to the remaining charge-power budget.

E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}} = \min\!\left(X_t, \frac{\max\!\left(S_{\min,t} - S_t^{(\mathrm{sd})}, 0\right)}{\eta_c}, \bar E_t^c\right)

E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}} = \min\!\left(\frac{\max\!\left(S_{\min,t} - S_t^{(\mathrm{sd,pv})}, 0\right)}{\eta_c}, \max\!\left(\bar E_t^c - E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}}, 0\right)\right)

Variables

E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}}, E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}}

gross reserve-restoration energy from PV and from grid

S_t^{(\mathrm{sd})}, S_t^{(\mathrm{sd,pv})}

SoC after self-discharge and after PV-backed reserve restoration

X_t, \bar E_t^c

remaining PV surplus and charge-power headroom

Interpretation

The model treats minimum SoC as a binding reserve floor. If self-discharge pushes the state of charge below it, that reserve is restored regardless of the electricity price.

(11)

Standby and auxiliary demand

Standby and auxiliary demand represents the ongoing electrical load of the battery, inverter, and control electronics within the interval. That power draw is first converted into an energy amount. The battery covers only the share that remains available above the minimum-SoC reserve after reserve restoration and within the discharge limit; any remainder is drawn directly from the grid.

E_t^{\mathrm{aux}} = P_{\mathrm{aux}} \Delta t

E_t^{\mathrm{aux}\leftarrow\mathrm{bat}} = \min\!\left(E_t^{\mathrm{aux}}, \eta_d \min\!\left(\max\!\left(S_t^{(\mathrm{reserve})} - S_{\min,t}, 0\right), \bar E_t^d\right)\right)

E_t^{\mathrm{aux}\leftarrow\mathrm{grid}} = E_t^{\mathrm{aux}} - E_t^{\mathrm{aux}\leftarrow\mathrm{bat}}

Variables

E_t^{\mathrm{aux}}

total auxiliary demand in interval t

E_t^{\mathrm{aux}\leftarrow\mathrm{bat}}, E_t^{\mathrm{aux}\leftarrow\mathrm{grid}}

auxiliary demand covered by the battery and by the grid

P_{\mathrm{aux}}, \bar E_t^d

auxiliary power and discharge-power limit

Interpretation

Auxiliary demand is therefore ordinary site consumption, not a separate battery-chemistry loss term. It reduces SoC only to the extent that withdrawable energy actually exists above the reserve floor; otherwise it increases grid import directly without triggering artificial grid-to-battery charging.

(12)

Economically admissible discharge

After direct PV self-consumption is allocated, the model reads the term from the inside out to determine how much energy the battery can physically deliver to load in this interval: first the energy available above the minimum reserve, then that amount capped by the interval discharge-power limit, and then converted with $\eta_d$ into the energy that actually reaches the load after discharge losses. Residual load is served only up to that amount and only if the current electricity price exceeds the discharge threshold.

E_t^{\mathrm{dis}\rightarrow\mathrm{load}} = \begin{cases}\min\!\left(L_t^{\mathrm{res}}, \eta_d \min\!\left(\tilde S_{t-1} - S_{\min,t}, \bar E_t^d\right)\right), & p_t > p_{\mathrm{discharge}}^{\min} \land p_t > p_{\mathrm{charge}}^{\max} \\ 0, & \text{otherwise}\end{cases}

Variables

E_t^{\mathrm{dis}\rightarrow\mathrm{load}}

battery energy delivered to load in interval t

L_t^{\mathrm{res}}

residual load after direct PV consumption

\tilde S_{t-1}, p_t

clipped start-of-interval SoC and current grid price

Interpretation

The nested min structure therefore reads in three stages: withdrawable energy above the reserve floor, the amount actually usable within the discharge limit, and finally no more than the current residual load. $\eta_d$ ensures that only the energy still available at the load after discharge losses is counted. The battery discharges only when doing so is both physically feasible and economically justified.

The additional condition $p_t > p_{\mathrm{charge}}^{\max}$ prevents discharge in price windows where charging would be economically preferred.

(13)

PV charging priority

PV surplus charges the battery before any possible grid charging. The limiting factors are surplus energy, free storage headroom, and charge power.

E_t^{\mathrm{pv}\rightarrow\mathrm{bat,gross}} = \min\!\left(X_t, \frac{S_{\max,t} - S_t^{(d)}}{\eta_c}, \bar E_t^c\right)

Variables

X_t

PV surplus after direct load coverage

S_t^{(d)}

SoC after the discharge decision and before PV charging

\bar E_t^c

maximum charge energy in the interval

Interpretation

In the implemented dispatch sequence, PV charging has priority over grid charging. Any surplus not absorbed by the battery is only handled afterwards via export or feed-in limitation.

(14)

Conditional grid charging

Grid charging is evaluated only after PV charging and can use only the remaining charge power in that interval. In addition, the current price must be below the charge threshold.

E_t^{\mathrm{grid}\rightarrow\mathrm{bat,gross}} = \begin{cases}\min\!\left(\frac{S_{\max,t} - S_t^{(\mathrm{pv})}}{\eta_c}, \max\!\left(\bar E_t^c - E_t^{\mathrm{pv}\rightarrow\mathrm{bat,gross}}, 0\right)\right), & p_t \le p_{\mathrm{charge}}^{\max} \\ 0, & \text{otherwise}\end{cases}

Variables

E_t^{\mathrm{grid}\rightarrow\mathrm{bat,gross}}

gross energy charged from the grid

S_t^{(\mathrm{pv})}

SoC after PV charging and before grid charging

p_{\mathrm{charge}}^{\max}

economic price threshold for grid charging

Interpretation

This prevents PV charging and grid charging from consuming the same charge-power budget twice. Grid charging remains restricted to intervals with sufficiently low prices.

(15)

Export and curtailment after charging

Any PV surplus left after battery charging is first treated as potential export and then, if required, capped by the active feed-in limit.

E_t^{\mathrm{exp,pre}} = \max\!\left(X_t - E_t^{\mathrm{pv}\rightarrow\mathrm{bat,gross}}, 0\right)

E_t^{\mathrm{exp}} = \min\!\left(E_t^{\mathrm{exp,pre}}, \phi_{\mathrm{fi}} P_{\mathrm{nom}}^{\mathrm{pv}} \Delta t\right)

E_t^{\mathrm{curt}} = E_t^{\mathrm{exp,pre}} - E_t^{\mathrm{exp}}

Variables

E_t^{\mathrm{exp,pre}}

potential export before limitation

E_t^{\mathrm{exp}}, E_t^{\mathrm{curt}}

actual export and curtailed energy

\phi_{\mathrm{fi}}, P_{\mathrm{nom}}^{\mathrm{pv}}

feed-in factor and nominal PV power

Interpretation

If no feed-in ceiling is active, $\phi_{\mathrm{fi}}$ is effectively 1 and curtailment disappears. Under an active cap, the difference is the modeled curtailed energy.

(16)

Grid import with battery

In the model, grid import consists not only of residual household demand but also of reserve-restoration charging from the grid, ordinary grid charging, and any uncovered auxiliary demand.

E_t^{\mathrm{grid,imp,load}} = \max\!\left(L_t^{\mathrm{res}} - E_t^{\mathrm{dis}\rightarrow\mathrm{load}}, 0\right)

E_t^{\mathrm{grid,imp}} = E_t^{\mathrm{grid,imp,load}} + E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}} + E_t^{\mathrm{grid}\rightarrow\mathrm{bat,gross}} + E_t^{\mathrm{aux}\leftarrow\mathrm{grid}}

Variables

E_t^{\mathrm{grid,imp,load}}

grid import used to serve household load

E_t^{\mathrm{grid,imp}}

total grid import in interval t

E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}}, E_t^{\mathrm{aux}\leftarrow\mathrm{grid}}

grid reserve top-up and auxiliary demand covered directly from the grid

Interpretation

This definition is central for cost and self-sufficiency metrics. Grid import rises whenever the battery is charged from the grid or when auxiliary demand cannot be covered from energy above the reserve floor.

(17)

State-of-charge update

The end-of-interval state of charge is obtained from the clipped starting state after subtracting self-discharge and auxiliary supply from the battery, and after adding reserve restoration as well as ordinary PV and grid charging.

S_t = \tilde S_{t-1} - E_t^{\mathrm{self}} + \eta_c E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}} + \eta_c E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}} - \frac{E_t^{\mathrm{aux}\leftarrow\mathrm{bat}}}{\eta_d} - \frac{E_t^{\mathrm{dis}\rightarrow\mathrm{load}}}{\eta_d} + \eta_c E_t^{\mathrm{pv}\rightarrow\mathrm{bat,gross}} + \eta_c E_t^{\mathrm{grid}\rightarrow\mathrm{bat,gross}}

Variables

\tilde S_{t-1}, S_t

clipped start-of-interval SoC and end-of-interval SoC

E_t^{\mathrm{self}}, E_t^{\mathrm{aux}\leftarrow\mathrm{bat}}

self-discharge loss and auxiliary demand supplied by the battery

E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}}, E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}}

reserve-restoration energy from PV and from grid

Interpretation

This equation ties the full dispatch sequence into one consistent terminal state for the interval. For SoC, charging counts only the energy that actually reaches the battery after charging losses. On discharge and on auxiliary demand served by the battery, SoC must fall by more than what later appears at the load, because additional discharge losses occur on the way. That is why charging is added only by the stored share, while discharge is subtracted by the full energy withdrawn from the battery.

Tariff and cost model

We separate fixed energy prices, dynamic interval prices, and annual fixed components. Price charts usually focus on variable EUR/kWh components; annual base charges are included in annual totals when configured.

(18)

Fixed tariff

Under fixed pricing, annual import cost is the energy price times import volume plus annual fixed components.

C_{\mathrm{fixed}} = p_{\mathrm{fixed}} E_{\mathrm{grid,imp}} + F_{\mathrm{annual}}

Variables

p_{\mathrm{fixed}}

flat energy price per kWh

E_{\mathrm{grid,imp}}

annual grid import

F_{\mathrm{annual}}

annual fixed tariff components

Interpretation

This is the reference path for flat tariffs and provides the baseline against which dynamic pricing logic can be compared.

(19)

Dynamic tariff

Under dynamic pricing, grid import is always priced on the chosen simulation time axis: hourly in hourly simulations and quarter-hourly in 15-minute simulations. Annual fixed charges are then added on top.

C_{\mathrm{dyn}} = \sum_t p_t^{\mathrm{dyn}} E_{\mathrm{grid,imp},t} + F_{\mathrm{annual}}

Variables

p_t^{\mathrm{dyn}}

time-varying price in interval t

E_{\mathrm{grid,imp},t}

grid import in interval t

F_{\mathrm{annual}}

annual base and network components

Interpretation

The dynamic-pricing path can be extended with regional network charges, levies, and supplier components; day-ahead prices come from the configured market data source, with DE-LU/AT sourced from Bundesnetzagentur | SMARD.de. For settlement, prices and energy flows are always brought onto the same simulation time axis. In 15-minute simulations, quarter-hourly prices are applied directly to the matching quarter-hourly flows. When energy flows are only available hourly, the model distributes each hourly energy amount evenly across the four quarter-hours in that path. When the simulation runs hourly, quarter-hourly prices are aggregated into hourly means and settled against the matching hourly flow. For rules that depend on negative spot prices, an hour in that path is already treated as affected when at least one quarter-hour within the hour is negative. Separately, the hourly price chart can also show hourly means plus min/max ranges derived from the quarter-hour source.

Reported metrics

We combine physical and economic performance metrics. The definitions below are the main outputs used for PV-only, battery-only, and combined-system comparison.

Comparison paths: uploaded data can already contain battery behavior. A scenario with 0 kWh of additional battery capacity is therefore not always identical to a pure household-without-battery baseline. Reported battery effects always refer to the matching comparison path.

(20)

Self-consumption rate

The self-consumption rate measures how much of total PV generation is used inside the system rather than exported.

\mathrm{SCR} = \frac{E_{\mathrm{pv,self}}}{E_{\mathrm{pv}}}

Variables

E_{\mathrm{pv,self}}

self-consumed PV energy

E_{\mathrm{pv}}

total PV generation

Interpretation

Higher values indicate that a larger share of generated solar electricity stays within the household or battery system.

(21)

Self-sufficiency

Self-sufficiency measures the share of household demand that can be covered without importing electricity from the grid.

\mathrm{SS} = 1 - \frac{E_{\mathrm{grid,imp}}}{E_{\mathrm{load}}}

Variables

E_{\mathrm{grid,imp}}

grid import over the evaluation period, including reserve-related and ordinary battery grid charging

E_{\mathrm{load}}

total household load

Interpretation

The lower residual grid import is relative to total demand, the higher the modeled energy independence. In the implementation path, grid import includes direct load import, reserve-restoration import from the grid, and gross grid charging of the battery.

This is an implementation-level import term, not only direct household import. That choice is deliberate: load later served from a battery that was charged from the grid does not count as genuine energy independence.

(22)

Battery interval net benefit

The battery’s economic contribution in each interval is the avoided import value minus the opportunity cost of all PV-backed charging, including reserve restoration, and minus the direct cost of all grid-backed charging and grid-supplied auxiliary demand.

B_t^{\mathrm{bat}} = V_t^{\mathrm{dis}} - OC_t^{\mathrm{pv}} - C_t^{\mathrm{grid}}

V_t^{\mathrm{dis}} = E_t^{\mathrm{dis}\rightarrow\mathrm{load}} \, p_t^{\mathrm{grid}}

OC_t^{\mathrm{pv}} = \left(E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}} + E_t^{\mathrm{pv}\rightarrow\mathrm{bat,gross}}\right) p_t^{\mathrm{feed\text{-}in}}

C_t^{\mathrm{grid}} = \left(E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}} + E_t^{\mathrm{grid}\rightarrow\mathrm{bat,gross}} + E_t^{\mathrm{aux}\leftarrow\mathrm{grid}}\right) p_t^{\mathrm{grid}}

Variables

B_t^{\mathrm{bat}}

net battery benefit in interval t

V_t^{\mathrm{dis}}, OC_t^{\mathrm{pv}}, C_t^{\mathrm{grid}}

discharge value, PV-charging opportunity cost, and direct grid-backed cost including auxiliary demand supplied from the grid

E_t^{\mathrm{reserve}\leftarrow\mathrm{pv,gross}}, E_t^{\mathrm{reserve}\leftarrow\mathrm{grid,gross}}

PV-backed and grid-backed reserve restoration flows plus direct auxiliary demand from the grid

Interpretation

In the current implementation path, any PV energy diverted into the battery is valued as foregone feed-in revenue, while any grid-backed battery charging enters as an explicit electricity purchase. Auxiliary demand served directly from the grid is also booked as an explicit negative battery contribution; auxiliary demand served from the battery is reflected indirectly through lower SoC and reduced later discharge potential. These interval values are aggregated over the simulation horizon and then annualized.

In country-specific special cases with netting or saldering logic, the final monetary effect can instead be derived scenario-wise from energy-cost differences.

(23)

Cumulative battery benefit

The interval-level battery benefit is first summed across all simulated intervals and only then annualized.

B_{\mathrm{bat,tot}} = \sum_t B_t^{\mathrm{bat}}

Variables

B_{\mathrm{bat,tot}}

cumulative battery benefit over the analysis period

B_t^{\mathrm{bat}}

battery benefit in interval t

Interpretation

This aggregation step turns interval battery contributions into the total amount from which annual battery savings are derived.

(24)

Battery cycles and expected life

The reported cycle count is based on total net charged energy relative to battery capacity. From that, an expected operating life in years is derived.

N_{\mathrm{cyc,tot}} = \frac{\sum_t E_t^{\mathrm{charge,net}}}{E_{\mathrm{cap}}}

N_{\mathrm{cyc,ann}} = \frac{N_{\mathrm{cyc,tot}}}{Y}

T_{\mathrm{life}} = \frac{N_{\mathrm{cyc}}}{N_{\mathrm{cyc,ann}}}

Variables

N_{\mathrm{cyc,tot}}, N_{\mathrm{cyc,ann}}

total and annual full-cycle equivalents

E_t^{\mathrm{charge,net}}, E_{\mathrm{cap}}

net charged energy in interval t and battery capacity

T_{\mathrm{life}}, N_{\mathrm{cyc}}

expected operating life and guaranteed cycles

Interpretation

The model uses full-cycle equivalents rather than literal start-stop cycles. This converts partial cycles into a consistent life-expectancy estimate.

(25)

Annual battery savings

Battery savings are the average annual benefit of the battery operation over the full analysis horizon.

C_{\mathrm{bat}}^{\mathrm{ann}} = \frac{B_{\mathrm{bat,tot}}}{Y}

Variables

B_{\mathrm{bat,tot}}

total battery benefit over Y years

Y

analysis horizon in years

Interpretation

This metric is the basis for annual battery ROI and simple battery payback.

(26)

Annual PV savings

Annual PV savings combine the value of self-consumed PV energy and export revenue and average them over the analysis horizon.

C_{\mathrm{pv}}^{\mathrm{ann}} = \frac{V_{\mathrm{pv,self}} + R_{\mathrm{feed\text{-}in}}}{Y}

Variables

V_{\mathrm{pv,self}}

value of self-consumed PV electricity

R_{\mathrm{feed\text{-}in}}

feed-in revenue

Interpretation

In total-system summaries this value is added to annual battery savings to derive total ROI, total profit, and total payback.

(27)

Battery ROI and payback

ROI and payback compress annual battery savings into two compact investment metrics.

\mathrm{ROI}_{\mathrm{bat}} = \frac{C_{\mathrm{bat}}^{\mathrm{ann}}}{I_{\mathrm{bat}}}\times 100

T_{\mathrm{pb,bat}} = \frac{I_{\mathrm{bat}}}{C_{\mathrm{bat}}^{\mathrm{ann}}}

Variables

C_{\mathrm{bat}}^{\mathrm{ann}}

annual battery savings

I_{\mathrm{bat}}

battery investment

T_{\mathrm{pb,bat}}

simple battery payback period

Interpretation

The implementation uses annualized savings. If annual savings are not positive, payback is represented by a large placeholder value rather than mathematical infinity.

These are simple annualized, non-discounted reporting metrics, not $\mathrm{NPV}$ or $\mathrm{IRR}$ .

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Total profit

Total profit is obtained by summing PV and battery savings across the analysis period and subtracting the relevant investments.

\Pi = Y\left(C_{\mathrm{bat}}^{\mathrm{ann}} + C_{\mathrm{pv}}^{\mathrm{ann}}\right) - I_{\mathrm{bat}} - I_{\mathrm{pv}}

Variables

\Pi

total system profit over the analysis period

C_{\mathrm{bat}}^{\mathrm{ann}}, C_{\mathrm{pv}}^{\mathrm{ann}}

annual battery and PV savings

I_{\mathrm{bat}}, I_{\mathrm{pv}}, Y

battery investment, PV investment, and analysis horizon

Interpretation

This is the profit term used when configurations are ranked by profit rather than ROI or self-sufficiency.

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Optimization objective

When multiple configurations are compared, the preferred configuration is the argmax of the chosen objective.

x_{\star}^{(g)} = \arg\max_x g(x)

g \in \{\mathrm{ROI}, \Pi, \mathrm{SS}\}

Variables

x

configuration consisting of PV size, battery capacity, and power

g

selected objective: battery ROI, total profit, or self-sufficiency

Interpretation

The best configuration for ROI does not have to be the same as the best configuration for total profit or self-sufficiency.

Economic conventions

The financial outputs follow a few deliberately simple conventions. These should be read together with the formulas so that ROI and payback are interpreted correctly.

Advanced finance metrics

NPV and IRR sit on top of the yearly nominal cash-flow model. Year 0 is upfront CAPEX; years 1 to Y are nominal net cash flows after maintenance.

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Net Present Value (NPV)

$\mathrm{NPV}$ discounts the yearly nominal net cash-flow stream back to year 0 and subtracts the upfront investment.

\mathrm{NPV} = -I_0 + \sum_{t=1}^{Y}\frac{CF_t^{\mathrm{nom}}}{(1+r)^t}

Variables

\mathrm{NPV}

discounted project value over the analysis horizon

I_0

upfront CAPEX in year 0

CF_t^{\mathrm{nom}}

nominal net cash flow after maintenance in year t

r, Y

discount rate and analysis horizon

Interpretation

A positive $\mathrm{NPV}$ means the project beats the chosen discount rate over the modeled horizon.

The implementation uses nominal yearly cash flows, with maintenance included in each yearly cash-flow term.

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Internal Rate of Return (IRR)

$\mathrm{IRR}$ is the discount rate that makes the same year-0 CAPEX and nominal yearly net cash-flow stream sum to zero.

0 = -I_0 + \sum_{t=1}^{Y}\frac{CF_t^{\mathrm{nom}}}{(1+\mathrm{IRR})^t}

Variables

\mathrm{IRR}

internal rate of return

I_0

upfront CAPEX in year 0

CF_t^{\mathrm{nom}}

nominal net cash flow after maintenance in year t

Y

analysis horizon

Interpretation

$\mathrm{IRR}$ can be compared with a hurdle rate or cost of capital. If the cash-flow pattern has no valid unique solution, the application reports N/A.

In the cumulative cash-flow chart, the dashed discounted line ends at NPV, while the solid nominal line shows the nominal payback path that underlies the headline payback metric. In representative-year finance plots, annual maintenance is allocated evenly across the displayed intervals so those visualizations stay consistent with the yearly cash-flow model.

Aspect	Current implementation	Interpretation
Analysis horizon	Multi-year simulation over the configured horizon, 20 years by default.	Reported annual figures are summaries across that horizon, not a standalone first-year simulation.
Inflation	The operational benefit path uses a configurable inflation assumption.	Multi-year totals therefore reflect nominal progression rather than a simple repetition of the initial year.
Average annual net cash flow	Displayed annual net cash flow equals the average yearly nominal cash flow implied by the accumulated multi-year result, after annual maintenance.	The number is a smoothed horizon average, not necessarily the cash flow of year one.
ROI and payback	Reported ROI uses average annual nominal net cash flow after maintenance divided by upfront investment; reported payback is the nominal break-even of the same maintenance-inclusive cash-flow series.	Both remain non-discounted headline metrics. Discounted metrics such as NPV and IRR are reported separately as advanced finance outputs.
NPV and IRR	NPV discounts year-0 CAPEX and yearly nominal net cash flows after maintenance at the configured discount rate. IRR is the rate that sets the same nominal cash-flow stream to zero.	Positive NPV means the project beats the chosen discount rate. IRR can be compared with a hurdle rate, but may be unavailable when the cash-flow stream has no valid unique solution.
Cumulative cash-flow chart	The plotted nominal line starts with upfront CAPEX and then adds yearly nominal cash flows net of maintenance; the dashed line discounts the same yearly cash flows.	The chart's zero crossing is the nominal cash-flow break-even of the plotted series and therefore matches the headline payback logic, apart from small rounding differences. Representative-year finance plots allocate maintenance evenly across intervals for visualization.

Country-specific special logic

The formulas above describe the generic dispatch and cost path. In some markets, additional regulatory logic can modify export treatment or final monetary attribution.

Base model

Without regulatory exceptions, dispatch, export, and monetary valuation follow the generic equations on this page. This base layer is identical across markets unless a country-specific override is triggered.

Override paths

When a market has an explicit special rule, it does not replace the entire model. It only overrides the affected sub-path, for example export limitation or final monetary attribution. The table below names those deviations explicitly.

Country	Rule	Model effect
Germany	Solarspitzengesetz path with possible feed-in limitation and zero compensation in negative-price intervals when the rule applies.	The generic export and revenue logic can therefore be constrained by country-specific regulation.
Netherlands	Saldering/netting logic can derive battery benefit from scenario-wise differences in net energy cost instead of the simple interval decomposition alone.	Final monetary impact can therefore follow regulatory cost logic rather than only the baseline interval-benefit formula.

Assumptions and current defaults

The values below are documented product defaults when users do not override them. They are model parameters, not physical constants.

Parameter	Current default	Note
Analysis horizon	20 years	Economic default; user overrides can replace it.
Time basis	Fixed-interval simulation at 15 or 60 minutes	Input series are normalized to a common 15-minute or 60-minute interval grid before dispatch, depending on the selected or detected dataset granularity.
SoC bounds	10% min / 95% max / 50% initial	Applied only when no explicit configuration is present.
Efficiencies	95% charge / 95% discharge	Affect both SoC updates and economic thresholds.
Self-discharge	3.0 %/month	Applied as a compounded interval loss on stored energy before reserve restoration.
Standby / auxiliary power	5 W	Modeled as direct electrical demand with battery-first coverage above minimum SoC and grid fallback for the remainder.
Guaranteed cycles	6000	Used in wear cost and life-expectancy estimates.
Battery chemistry	LiFePO4 default; catalog chemistry for real models; generic Li-ion treated conservatively as NMC	Chemistry determines the calendar- and cycle-aging coefficients. If no narrower chemistry information is available, LiFePO4 remains the default for manual configurations.
Battery aging	Chemistry-aware calendar and cycle aging; LiFePO4 default	Manual and freely configured batteries default to LiFePO4; real battery models use catalog chemistry when available.
Modeled aging stressors	Average SoC, cycle depth, and guaranteed cycles	Temperature, explicit C-rate, and manufacturer-specific cell or thermal models are not yet modeled separately.
Aging sensitivity controls	Calendar 1.0x and cycle 1.0x by default	Optional user-facing multipliers scale the baseline calendar- and cycle-aging terms for sensitivity analysis, while the literature-based model structure stays fixed.
PV degradation	0.5% per year	Default used in multi-year PV projections.
Multi-year replay of uploaded profiles	Without an existing battery, the measured hourly PV→load overlap is preserved	For uploads without an existing battery, we reconstruct hourly direct PV usage from PV generation, grid import, and grid export and carry that overlap structure into later years. This keeps hours visible in which strong PV output still leaves residual load peaks. If later years were rebuilt only from min(PV, load), those peaks would disappear and the additional battery benefit would tend to be understated. PV degradation still reduces the available PV energy over time.
Uploads with an existing battery	Observed battery behaviour is replayed approximately; the uploaded battery is not synthetically aged	When uploaded profiles already contain battery charge and discharge flows, we do not infer a separate technical model for that battery in the multi-year path. Instead, we carry its observed behaviour forward approximately. PV degradation still reduces available PV energy, so PV-backed portions remain feasible only to the extent that degraded PV still allows. Long-horizon results with an uploaded existing battery should therefore be read as approximations.
New-system scenario construction	Each candidate PV size gets its own full-year PV profile; battery variants are simulated on the same demand and tariff basis	In the new-system workflow, the demand baseline comes from uploaded or generated consumption data. For each candidate PV size, we first assign available PV capacity to the roof surfaces with the highest expected yield and send only those allocated partial surfaces to the PV profile calculation. This creates a full-year PV profile; we then simulate each PV/battery combination against the same horizon and optimization objective. Manual batteries use the configured assumptions; real battery mode uses catalog specifications where available.
Annual fixed tariff components	Included in annual totals when configured	EUR/kWh charts often show only variable price components.

Data sources and model inputs

The methodology combines user-supplied inputs with external data sources. The table below lists the main sources and their role in the model.

Source	Used for	Notes
User inputs and uploads	Household consumption, existing PV data, battery assumptions, price assumptions	Uploaded time series take precedence over coarse default profiles where available.
PVGIS 3-year profile	Site-specific hourly PV production values from three real historical PVGIS years	PVGIS calculates production from coordinates, orientation, tilt, and installed capacity; we use 14% system losses and take the middle-yield month for each calendar month. This preserves real weather sequences without modelling a full uncertainty range.
Bundesnetzagentur \| SMARD.de	Day-ahead spot prices for DE-LU/AT dynamic pricing	Used in quarter-hour resolution. Source attribution: Bundesnetzagentur \| SMARD.de, licensed under CC BY 4.0. Voltary processes the source data (including unit conversion, interval alignment, and chart aggregation).
Tariff and network-charge inputs	Regional network fees, levies, supplier margins, country-specific price components	Included through configuration, operator-specific parameters, and pricing-model logic.
Regulatory inputs	Special rules such as negative-price handling or feed-in limitation	Applied only when country, installation date, and rule path require it.

Methodological references

The structure of the aging model and the key modeling assumptions are anchored in the following references. The concrete coefficients and multipliers are still initial calibration values within those literature envelopes, not untouched copies of any single paper.

Topic	Methodological contribution	Source
SAM / NREL battery life	Supports the separation of calendar and cycle aging and DoD-dependent cycle logic.	NREL SAM battery life model overview
Lifetime-prediction review	Supports separating calendar and cycle contributions as a robust model structure.	Battery lifetime prediction review
LFP calendar aging	Supports SoC-dependent calendar aging and low-to-moderate baseline fade for LFP.	LFP storage-aging study
Square-root-of-time calendar aging	Supports the square-root-of-time form for calendar aging and the strong impact of high SoC levels.	Automotive aging study
LFP cycle-life model	Supports using cycle depth as a central driver of cycle aging.	LFP cycle-life model
LFP vs NMC benchmark context	Provides benchmark context for conservative chemistry assumptions and relative life-expectancy ordering.	PNNL / Sandia benchmark

Short worked example

The example below is a deliberately simplified single-interval calculation. It illustrates the battery decision sequence and the economic interpretation without carrying all multi-year effects such as degradation or annual fixed charges.

Step	Calculation	Result	Interpretation
1. Inputs	Load 4.0 kWh; PV 1.0 kWh; starting SoC 6.0 kWh; capacity 10.0 kWh; SoC bounds 10% and 95%; P_d = 3.0 kW; η_d = 0.95; price 0.30 EUR/kWh.	Complete interval state before dispatch.	This example isolates one hourly interval; degradation and annual fixed charges are intentionally excluded.
2. Direct coverage and residual load	PV→load = min(1.0, 4.0) = 1.0. Therefore L^res = 4.0 - 1.0 = 3.0.	Direct PV coverage 1.0 kWh; residual load 3.0 kWh.	The battery only decides after this direct PV allocation has been made.
3. Discharge and grid import	Maximum deliverable to load: 0.95 × min(6.0 - 1.0, 3.0) = 2.85. Remaining grid import: 3.0 - 2.85 = 0.15.	Battery discharge 2.85 kWh; grid import 0.15 kWh.	Usable discharge is limited by internal reserve above S_min and by discharge efficiency.
4. Updated SoC and interval benefit	S_t = 6.0 - 2.85 / 0.95 = 3.0. Interval benefit: 2.85 × 0.30 = 0.855 EUR.	End-of-interval SoC 3.0 kWh; interval monetary value 0.855 EUR.	Because there is neither PV opportunity cost nor grid charging in this example, the interval benefit equals the avoided import value.
5. Annual metric	With average annual battery savings of 720 EUR and battery investment of 6,000 EUR, ROI = 720 / 6000 × 100 and T_pb = 6000 / 720.	Annualized ROI 12%; simple payback 8.3 years.	These are annualized, non-discounted metrics and therefore not NPV/IRR measures.

Model limitations

Future electricity prices, levies, and regulation are uncertain. The simulation is therefore a structured scenario analysis, not a price guarantee.
Current outputs are point estimates under the chosen assumptions, not stochastic ranges, confidence intervals, or uncertainty bands.
Usable battery capacity ages over the multi-year path through chemistry-aware calendar and cycle components. Temperature, charge/discharge rate, and manufacturer-specific cell or thermal models are not yet represented separately.
If an upload already contains battery flows, we currently treat that existing battery as observed behaviour in the multi-year path and do not synthetically age it. PV-backed portions are only carried forward insofar as degraded PV still makes them feasible. Long-horizon projections with an uploaded existing battery therefore remain approximate unless that battery is modeled separately with its own technical parameters.
Household behaviour changes, manual intervention, and manufacturer-specific control strategies are only captured to the extent that they are represented in the inputs and assumptions.
Installer quotes, financing costs, tax edge cases, and project-specific ancillary costs are outside the baseline equations documented here.
Advanced finance outputs are pre-tax and unlevered. Taxes, grants, depreciation, salvage value, and replacement logic are intentionally excluded from the current DCF layer.
The outputs are most reliable for comparing configurations under consistent assumptions. Realized absolute values can still differ later.

Questions about an assumption or equation?

If you want to discuss an edge case, refer directly to the metric or equation number. That makes it easier to trace the relevant calculation path precisely.

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