Battery Storage¶

MicroGridsPy models the battery energy storage system as an aggregated electrochemical device described through charging, discharging, and state-of-charge (SOC) dynamics. The model combines three layers:

power limits, which bound charging and discharging rates;
energy-balance constraints, which govern the evolution of stored energy;
usable-capacity constraints, which limit operation according to nominal or degraded available energy.

Two representations are supported: a constant-efficiency special case, and an advanced convex-loss formulation where AC-side efficiency varies with power. In the typical-year formulation the battery is a single block operated cyclically over a representative year; in the multi-year formulation, storage investment is cohort-based, enabling cohort availability and inter-year state propagation. Semi-empirical degradation is available in both — priced through a binding-life CAPEX amortisation in the typical-year model, and additionally tracked as a capacity-fade state in the multi-year model (see Battery degradation). At hourly resolution \(\Delta t = 1\,\text{h}\), so power (kW) and one-hour energy transfers (kWh) are interchangeable in the storage balance.

Constant-efficiency dispatch¶

The simplest representation uses fixed charge/discharge efficiencies \(\eta_{\text{ch}}, \eta_{\text{dis}}\).

Typical-year formulation¶

Charging and discharging are bounded by the installed battery inverter (converter) power \(P^{\text{inv}}\), an explicit sizing variable that may optionally be coupled to the installed energy through a maximum C-rate \(c^{\text{rate}}\) (\(P^{\text{inv}} \le c^{\text{rate}}\,C^{\text{bat}}\)):

\[ P^{\text{ch}}_{t,\omega} \le P^{\text{inv}}, \qquad P^{\text{dis}}_{t,\omega} \le P^{\text{inv}} \qquad \forall t,\omega \]

The state of charge evolves as

\[ \begin{aligned} \text{SOC}_{0,\omega} &= \text{SOC}_0\, C^{\text{bat}} \\[3pt] \text{SOC}_{t,\omega} &= \text{SOC}_{t-1,\omega} + \eta_{\text{ch}} P^{\text{ch}}_{t-1,\omega} - \frac{P^{\text{dis}}_{t-1,\omega}}{\eta_{\text{dis}}} \qquad \forall t=1,\dots,|\mathcal{T}|-1 \end{aligned} \]

A cyclic boundary condition closes the representative year:

\[ \text{SOC}_{|\mathcal{T}|-1,\omega} + \eta_{\text{ch}} P^{\text{ch}}_{|\mathcal{T}|-1,\omega} - \frac{P^{\text{dis}}_{|\mathcal{T}|-1,\omega}}{\eta_{\text{dis}}} = \text{SOC}_0\, C^{\text{bat}} \qquad \forall\omega \]

and the SOC is bounded by the depth-of-discharge limit:

\[ (1-\text{DoD})\, C^{\text{bat}} \le \text{SOC}_{t,\omega} \le C^{\text{bat}} \qquad \forall t,\omega \]

Multi-year formulation¶

For each year \(y\), scenario \(\omega\), and cohort \(k\), charge/discharge power is bounded by the active installed inverter power \(P^{\text{inv}}\), while the SOC is bounded by the available cohort energy capacity \(\overline{C}^{\text{bat}}_{y,k}\):

\[ P^{\text{ch}}_{t,y,\omega,k} \le P^{\text{inv}}, \qquad P^{\text{dis}}_{t,y,\omega,k} \le P^{\text{inv}} \]

\[ (1-\text{DoD})\,\overline{C}^{\text{bat}}_{y,k} \le \text{SOC}_{t,y,\omega,k} \le \overline{C}^{\text{bat}}_{y,k} \]

Within each year, SOC follows the same constant-efficiency recursion, indexed by \((y,\omega,k)\). A key difference from the typical-year case is that the multi-year model does not impose a cyclic yearly boundary: the terminal state of one year is carried forward as the initial state of the next, unless a cohort is newly installed or replaced.

Advanced battery loss model¶

The advanced model replaces the constant one-way efficiencies with an explicit representation of power-dependent conversion losses, recast as an LP-safe convex piecewise-linear epigraph.

The public AC-side variables (\(P^{\text{ch}}, P^{\text{dis}}\), and the stored energy SOC) are complemented by internal DC-side variables: the DC power actually stored/withdrawn (\(P^{\text{ch,dc}}, P^{\text{dis,dc}}\)) and the conversion losses \(L^{\text{ch}}, L^{\text{dis}}\). The AC/DC balances are

\[ P^{\text{ch}} = P^{\text{ch,dc}} + L^{\text{ch}}, \qquad P^{\text{dis}} = P^{\text{dis,dc}} - L^{\text{dis}} \]

so when charging, the AC power drawn exceeds the energy stored, and when discharging, the AC power delivered is lower than the internal energy withdrawn. The curve is normalized on the installed inverter power, so both reference powers equal the inverter design variable, \(P^{\text{ref,ch}} = P^{\text{ref,dis}} = P^{\text{inv}}\), and the losses satisfy epigraph constraints for each interpolation segment \(i\):

\[ \begin{aligned} L^{\text{ch}}_{t,\omega} &\ge m^{\text{ch}}_i\,P^{\text{ch,dc}}_{t,\omega} + q^{\text{ch}}_i\,P^{\text{ref,ch}} \\[3pt] L^{\text{dis}}_{t,\omega} &\ge m^{\text{dis}}_i\,P^{\text{dis,dc}}_{t,\omega} + q^{\text{dis}}_i\,P^{\text{ref,dis}} \end{aligned} \qquad \forall i,t,\omega \]

The coefficients \(m_i, q_i\) are precomputed from the battery efficiency-curve input, keeping the optimization linear and tractable. The curve is fit through the user's sampled operating points only — it is not forced through the origin — so the lowest segment can carry a positive intercept \(q_i\,P^{\text{inv}}\) representing a no-load / standby loss. This is what lets the model express a physically realistic peaked efficiency: efficiency is best near an intermediate power and falls at both very low power (the fixed standby loss dominates a small throughput) and full power (conversion and resistive losses grow). A curve anchored at the origin would have \(L(0)=0\) and could only ever be monotonically more efficient at low power, which would make the power-dependent model uniformly cheaper than the constant baseline. The system boundary is AC-to-AC (the inverter/converter plus the cell path), so the one-way efficiencies represent the full round-trip conversion of a modern LFP + hybrid-inverter system.

Under the advanced model the SOC balance uses the DC-side flows,

\[ \text{SOC}_{t,\omega} = \text{SOC}_{t-1,\omega} + P^{\text{ch,dc}}_{t-1,\omega} - P^{\text{dis,dc}}_{t-1,\omega} \]

so the constant-efficiency recursion is a special case: when the advanced model is active, \(\eta_{\text{ch}}, \eta_{\text{dis}}\) are no longer used directly and efficiency is captured through explicit losses.

Battery losses: left, a convex loss curve with a positive no-load intercept approximated by piecewise-linear segments; right, the resulting peaked charge and discharge efficiency versus relative DC power

Power-dependent battery losses and the resulting one-way efficiency. Left: the convex loss curve, with a positive no-load (standby) intercept, approximated through piecewise-linear segments. Right: the corresponding charge and discharge efficiency — peaked at an intermediate power and lower at both very low power (standby-dominated) and full power (conversion/resistive losses).

Battery degradation¶

Battery ageing is represented with a linearised semi-empirical model. The idea (after Andrade's decomposition) is to collapse a physics-grade electro-thermal + ageing simulation into a single linear recursion on usable energy capacity,

\[ E_t = E_{t-1} \;-\; \alpha\, E^{\text{B}} \;-\; \beta\, P^{\text{BE}}_t \]

where \(E^{\text{B}}\) is the nameplate energy, \(P^{\text{BE}}_t\) the energy exchanged in hour \(t\), \(\alpha\) the calendar coefficient (always active, time-based) and \(\beta\) the cycle coefficient (active only when power flows, use-based). The strength of the approach is that \(\alpha\) and \(\beta\) are pre-fitted, exogenous coefficients: they already carry the dominant stress factors — temperature, depth-of-discharge and chemistry — yet the optimisation sees them as constants, so it stays linear.

Semi-empirical coefficients \(\alpha(T)\) and \(\beta(T)\)¶

Both coefficients are cubic polynomials in a scaled ambient temperature \(y = T_{\text{env}}/10\):

\[ \alpha_{\text{hour}}(T) = c_1 y^3 + c_2 y^2 + c_3 y + c_4, \qquad \beta_{\text{hour}}(T) = \big(d_1 y^3 + d_2 y^2 + d_3 y + d_4\big)\cdot \frac{N_{\text{ref}}}{N_{\text{user}}} \]

The shape (the fitted \(c_i, d_i\)) is built in, selected by chemistry ∈ {LFP, NMC, lead-acid} and by stress band: \(\beta\) per DoD band (50–90 % for Li-ion), \(\alpha\) per SoC band chosen from the DoD (deep discharge → high-SoC storage → 20 % band; shallow → 40 % band). Lead-acid uses a single temperature-independent \(\beta\) in \(z = 10\,\text{DoD} - 2\). The magnitude is user-scalable through a parameter you already provide — the rated cycle life \(N_{\text{user}}\) — via the ratio \(N_{\text{ref}}/N_{\text{user}}\), which ports the validated curve to a battery of any cycle life without re-fitting. This is the same "fixed shape × user scale" pattern used by the generator partial-load curve.

Semi-empirical degradation coefficients: beta rising with temperature for LFP DoD bands and NMC (left), and the annual calendar rate alpha rising with temperature by chemistry and SoC band (right)

Semi-empirical coefficients versus ambient temperature. Left: the cycle coefficient \(\beta(T)\) — rising with temperature, and depth- and chemistry-dependent. Right: the calendar coefficient expressed as an annual rate \(\alpha(\bar T)\cdot 8760\) — rising with temperature and higher at higher average SoC. Both are evaluated offline, so the optimisation stays linear.

These are driven by two inputs, required only when cycle-fade degradation is enabled:

ambient_temperature.csv — an hourly ambient-temperature series (°C), with the same scenario/year layout as load_demand.csv;
battery.technical.chemistry — one of LFP, NMC, lead_acid;

together with battery.technical.depth_of_discharge (sets the band), the rated cycle_lifetime_to_eol_cycles, and the state-of-health span initial_soh / end_of_life_soh.

Typical-year: amortisation over the binding life¶

The typical-year model has no multi-year capacity state, so degradation is purely an economic effect. It must stay consistent with the objective's annuity convention: every asset's CAPEX is turned into a level annual charge \(\text{CRF}(\text{wacc}, L)\cdot\text{CAPEX}\) (cost of capital included) and paid every year, which already prices replacement over the calendar life — so degradation acts on the effective lifetime, not as a separate charge. The battery energy CAPEX is therefore recovered by the larger of a calendar annuity and a cycle annuity (i.e. amortised over the shorter of the two lives), via a per-scenario epigraph \(Z_\omega\):

\[ \begin{aligned} Z_\omega &\ge \text{CRF}(\text{wacc}, L^{\text{cal}})\,\text{CAPEX}\; C^{\text{bat}} && \text{(calendar annuity)}\\[3pt] Z_\omega &\ge c^{\text{repl}}\,\varphi \sum_t \beta_{t,\omega}\big(P^{\text{ch}}_{t,\omega}+P^{\text{dis}}_{t,\omega}\big) && \text{(cycle annuity)} \end{aligned} \qquad c^{\text{repl}} = \frac{\text{CAPEX}}{\text{SoH}_0-\text{SoH}_{\text{eol}}},\;\; \varphi = \text{CRF}(\text{wacc}, L^{\text{cal}})\,L^{\text{cal}} \]

Minimising \(\sum_\omega w_\omega Z_\omega\) drives \(Z\) to the maximum of the two, so the battery is charged CAPEX ÷ min(calendar, cycle-limited) life. When cycling is gentle the calendar limit binds and degradation adds nothing; when it is hard enough to shorten the life below the calendar value the cycle limit binds and the cost rises. Hotter operation raises \(\beta(T)\) and so brings the crossover forward. This mirrors the multi-year treatment below (same \(\varphi\) calibration and the same flat-then-rising cost envelope shown in the replacement-cost figure) but needs neither a capacity state nor the convex-loss model.

Multi-year: capacity-fade state¶

The dynamic multi-year model tracks an explicit yearly usable-capacity state per scenario and investment cohort. The available nominal capacity of a cohort is

\[ \overline{C}^{\text{bat}}_{y,k} = u_k\, C^{\text{bat}}_{\text{nom}}\, a_{y,k}\, g_{y,k}, \]

with \(u_k\) units, activity mask \(a_{y,k}\), and a calendar-fade factor \(g_{y,k}\) (below). The effective usable capacity \(C^{\text{eff}}_{y,\omega,k}\le \overline{C}^{\text{bat}}_{y,k}\) is the energy left after cycle fade; it is constant within a year and evolves across years.

Cycle fade is accumulated per year from the DC-side energy exchanged. The representation is selected by chemistry, not by a user switch:

Li-ion (LFP, NMC) — depth-resolved marginal bands. The usable SOC window is split into \(K\) equal bands; discharge is routed through a stacked band reservoir and each band \(b\) carries a convex marginal wear cost \(c_b(T)\) that increases with depth. Annual fade is

\[ F^{\text{cyc}}_{y,\omega,k} = \sum_t \sum_{b=1}^{K} c_{b}(T_{t,y})\; P^{\text{dis,dc}}_{b,t,y,\omega,k} \]

Because deeper bands cost more, the optimiser fills shallow bands first, so the cycling depth is emergent — the model discovers how deep to cycle rather than assuming a fixed DoD — while the block stays a pure LP. The band marginals \(c_b = \partial\Psi/\partial D\) are distilled from an offline physics-grade depth curve \(\Psi(D,T)\) (archived separately — see the references), re-binned to \(K\) usable bands and scaled by the same \(N_{\text{ref}}/N_{\text{user}}\) cycle-life ratio as \(\beta\). The band count \(K\) is set by battery_model.degradation_model.n_soc_bands (default 5). A full-depth cycle reproduces the same per-cycle fade as the flat coefficient, so the two representations agree in the aggregate and differ only in how they price partial-depth cycling.

Lead-acid — flat \(\beta(T)\). The depth-band model is Li-ion only, so lead-acid keeps the single semi-empirical coefficient \(\beta(T)\) times the DC-side throughput (both directions, matching \(\beta\)'s calibration of \(2\,\text{DoD}\) per full cycle):

\[ F^{\text{cyc}}_{y,\omega,k} = \sum_t \beta_{t,y,\omega}\, \big(P^{\text{ch,dc}}_{t,y,\omega,k} + P^{\text{dis,dc}}_{t,y,\omega,k}\big) \]

Calendar fade is applied through the exogenous factor \(g_{y,k}\), whose rate is driven by \(\alpha(T)\) rather than a flat user %/yr:

\[ r^{\text{cal}}_{y} = \alpha_{\text{hour}}(\bar T_y)\cdot 8760, \qquad g_{y,k} = \prod_{j=\text{commission}}^{\,y-1}\big(1 - r^{\text{cal}}_{j}\big) \]

where \(\bar T_y\) is the annual-mean ambient temperature. The product accumulates over the years a cohort has lived and restarts at each replacement; for a constant temperature it reduces to \((1-r^{\text{cal}})^{\text{age}-1}\).

The state evolves by carrying capacity forward minus that year's cycle fade, resetting a newly commissioned cohort to \(C^{\text{reset}}_{y,k} = \text{SoH}_0\,\overline{C}^{\text{bat}}_{y,k}\):

\[ C^{\text{eff}}_{y,\omega,k} \;\;\{\le\;\text{or}\;=\}\;\; \big(C^{\text{eff}}_{y-1,\omega,k} - F^{\text{cyc}}_{y-1,\omega,k}\big)\,(1-b_{y,k}) + C^{\text{reset}}_{y,k}\, b_{y,k} \]

The link is an exact equality when the availability ceiling does not decline (no calendar rate), and an inequality (take the smaller of the carried state and the declining ceiling) otherwise. The effective capacity then limits the stored energy, so fade shrinks the usable SOC window while charge/discharge power stays bounded by the inverter:

\[ (1-\text{DoD})\,C^{\text{eff}}_{y,\omega,k} \le \text{SOC}_{t,y,\omega,k} \le C^{\text{eff}}_{y,\omega,k} \]

The reported state of health \(\text{SoH} = C^{\text{eff}}/\overline{C}^{\text{bat}}\) is reconstructed from this physical recursion.

Multi-year: replacement economics — amortisation over the binding life¶

MicroGridsPy costs every asset as a level annuity (CRF × CAPEX paid each active year, cost of capital included), which is a pay-as-you-go rental: it needs no salvage term and already embeds replacement every calendar lifetime. Degradation must therefore act on the effective lifetime — not as a second, parallel wear charge (which would double-count the replacement already priced by the annuity). The battery energy CAPEX is recovered by amortising it over the binding life, i.e. the shorter of the calendar life and the cycle-limited life. This is written as a linear max-of-two-annuities epigraph on a per-cohort cost \(Z_{y,\omega,k}\):

\[ \begin{aligned} Z_{y,\omega,k} &\ge \underbrace{u_k\,C^{\text{bat}}_{\text{nom}}\,\text{CAPEX}\,\cdot \text{CRF}(\text{wacc}, L^{\text{cal}})\, a_{y,k}}_{\text{calendar annuity}} \\[4pt] Z_{y,\omega,k} &\ge \underbrace{c^{\text{repl}}\,\varphi\; F^{\text{cyc}}_{y,\omega,k}}_{\text{cycle annuity}}, \qquad \varphi = \text{CRF}(\text{wacc}, L^{\text{cal}})\cdot L^{\text{cal}} \end{aligned} \]

Minimising \(\sum_y \text{disc}_y \sum_\omega w_\omega \sum_k Z_{y,\omega,k}\) drives \(Z\) to the maximum of the two, which equals the CAPEX amortised over \(\min(L^{\text{cal}}, L^{\text{cyc}})\) — the effective (economic = technical) lifetime. The financing gross-up \(\varphi\) is calibrated so the two bounds meet exactly at the crossover (cycle life = calendar life), making the switch continuous; \(\varphi = 1\) when WACC = 0. The battery inverter CAPEX stays calendar-amortised.

Battery CAPEX amortised over the binding life: a flat calendar annuity, a rising cycle annuity, and their maximum forming a flat-then-rising envelope with a kink at the crossover

Amortising battery energy CAPEX over the binding life. Below the crossover (gentle cycling) the calendar limit binds and the charged cost is flat — degradation adds nothing beyond what the ordinary annuity already prices. Above it (hard cycling) the cycle limit binds and the cost rises, so cycling is penalised only when it actually shortens the battery's life. The optimiser can respond by oversizing storage to cycle each unit more gently and push the cycle life back toward the calendar limit.

Configuration & inputs¶

Switch	Behaviour
Efficiency model	constant round-trip efficiency, or power-dependent convex losses (no ageing on its own)
Cycle fade (`cycle_fade_enabled`)	Semi-empirical throughput wear. Typical-year: binding-life CAPEX amortisation from \(\beta(T)\). Multi-year: a capacity-fade state, using depth-resolved per-SOC-band marginals \(c_b(T)\) for Li-ion (LFP/NMC; `n_soc_bands` bands, default 5) and the flat \(\beta(T)\) for lead-acid
Calendar fade	in multi-year, the \(\alpha(T)\)-driven annual rate \(r^{\text{cal}}_y\) on the availability factor

Endogenous cycle fade in the multi-year model requires the convex-loss efficiency model (throughput is defined on the internal DC-side powers); the typical-year cost does not. When cycle fade is enabled, provide ambient_temperature.csv, battery.technical.chemistry, cycle_lifetime_to_eol_cycles, and end_of_life_soh.

Future extensions

Discrete state-of-health-triggered replacement (replace exactly when SoH reaches end-of-life) is the rigorous form of the binding-life idea, but it introduces integer replacement decisions and a salvage term, so the linear max-of-annuities is used instead. Other possible additions: power fade / SoH-dependent power derating, and representative-day clustering to speed up the full 8760 × Y degradation solve.

References¶

The decomposition and the fitted \(\alpha\)/\(\beta\) coefficients follow S. Andrade, Battery Degradation Modelling for Off-Grid Energy System Sizing: Methodology and Case Study in the African Context (MSc dissertation, Politecnico di Milano / FCUL, 2023), which calibrates the linear recursion against a physics-based electro-thermal + semi-empirical reference model.

The Li-ion depth-resolved per-SOC-band marginals \(c_b(T)\) are distilled from an offline Layer-I physical model (electro-thermal + semi-empirical ageing), which computes the depth curve \(\Psi(D,T)\) directly. That model and its distilled coefficient file are archived separately; see the project's data archive for the DOI.