Objective Function¶

Both planning modes minimize an expected total system cost built from annuitized investment costs, expected operating costs, and optional externalities. The two modes share the same economic logic but differ in how time is represented.

All monetary quantities are expressed in real (inflation-free) terms, measured in today's money. Prices and costs are net of general inflation, and intertemporal discounting uses real discount rates, so values at different times are directly comparable.

Multi-Year Planning¶

The multi-year mode minimizes the discounted expected system cost incurred within the modelled horizon, accounting for annualized investment payments, expected operating costs, externality costs, and embodied-emission costs, all in present value:

\[ \min \; \sum_{y=1}^{H} \frac{SC_y}{(1+r_s)^{y}} \]

where \(r_s\) is the social discount rate and the annual system cost \(SC_y\) is

\[ SC_y = \text{Annuity}_y + \sum_{\omega\in\Omega} p_\omega \left( \text{O}\&\text{M}_{y,\omega} + \text{Externalities}_{y,\omega} \right) + \text{EmbodiedEmissionCost}_y , \]

with \(\omega\in\Omega\) the scenarios, each of probability \(p_\omega\). The three cost blocks decompose as

\[ \begin{aligned} \text{Annuity}_y &= \sum_{j,k} \alpha_{j,k,y}\,\text{CAPEX}_{j,k}\,\text{CRF}_{j,k} \\[4pt] \text{O}\&\text{M}_{y,\omega} &= \text{FixedO}\&\text{M}_{y,\omega} + \text{FuelCost}_{y,\omega} + \text{GridImportCost}_{y,\omega} - \text{GridExportRevenue}_{y,\omega} - \text{Subsidy}_{y,\omega} \\[4pt] \text{Externalities}_{y,\omega} &= \text{LLCost}_{y,\omega} + \text{DirectEmissionCost}_{y,\omega} + \text{GridEmissionCost}_{y,\omega} \end{aligned} \]

where \(\alpha_{j,k,y}\) is an activation mask that switches on the contribution of cohort \(k\) of technology \(j\) from its commissioning step onward.

Annuities and the capital recovery factor¶

Investment costs use an annuity-based accounting framework with explicit investment steps. Each cohort of technology \(j\), commissioned at step \(\tau\) with present investment cost \(I_{j,\tau}\), technical lifetime \(LT_j\), and weighted average cost of capital \(\text{WACC}_{j,\tau}\), is converted into a stream of constant annual payments through the capital recovery factor (CRF):

\[ \text{CRF}_{j,\tau} = \frac{\text{WACC}_{j,\tau}\,(1+\text{WACC}_{j,\tau})^{LT_j}} {(1+\text{WACC}_{j,\tau})^{LT_j}-1} \]

The annuities are activated from the commissioning year and, in the current implementation, persist over the remaining horizon through an implicit like-for-like replacement logic. This represents phased investment and delayed deployment while avoiding explicit reinvestment variables.

Technical vs. economic lifetime

The annuity formulation implicitly spreads capital repayment over the technical lifetime; no distinction is made between technical and economic lifetime. Cohort annuity streams remain active over the remaining modelled horizon via an implicit replacement convention. This ensures internal consistency and avoids explicit reinvestment variables, but differs from a full-upfront-CAPEX accounting framework.

Weighted Average Cost of Capital¶

The WACC represents the opportunity cost of capital:

\[ \text{WACC}_j = \frac{E_j}{E_j + D_j}\,K^E_j + \frac{D_j}{E_j + D_j}\,K^D_j\,(1-T) \]

where \(E_j\) and \(D_j\) are equity and debt shares, \(K^E_j\) and \(K^D_j\) the costs of equity and debt, and \(T\) the corporate tax rate. In MicroGridsPy the WACC is an explicit, configurable parameter, able to represent concessional finance, public-sector investment, or policy-driven de-risking.

Dual-rate discounting¶

Intertemporal evaluation follows a dual-rate logic. Capital recovery for each technology uses its WACC, while all system-level cash flows entering the objective — annuities, operating costs, externalities, embodied-emission costs — are discounted to present value using a social discount rate \(r_s\). This separates financial opportunity costs at the asset level from societal time preferences at the system level. Following the Ramsey formulation:

\[ r_s = \rho + \eta g \]

where \(\rho\) is the pure rate of time preference, \(\eta\) the elasticity of marginal utility of consumption, and \(g\) the expected long-term growth of per-capita consumption.

Time horizon, end-of-horizon bias, and accounting conventions¶

Long-term capacity expansion is conceptually an infinite-horizon problem, but optimization is performed over a finite modelled horizon. This truncation can introduce end-of-horizon bias, where investments near the terminal year are mis-valued if their remaining lifetime is not accounted for — particularly for long-lived, capital-intensive technologies.

How horizon-end effects are treated depends on the accounting convention:

Under a full-upfront-CAPEX convention, the entire investment is charged at commissioning and a salvage value correction is required for useful life beyond the horizon.
Under an annuity-based convention, investment costs are annualized and only payments within the modelled years are counted, which naturally mitigates the most severe truncation effects.

MicroGridsPy's multi-year objective follows the annuity-based convention: only annualized payments falling within the horizon enter the objective. A salvage-related quantity may still be computed in post-processing for reporting, but it does not enter the optimization. The formulation should therefore be read as a finite-horizon approximation of a long-term planning problem, not one fully immune to terminal-horizon distortions.

Typical-Year Planning¶

The typical-year mode minimizes the expected equivalent annual cost (EAC). Investment decisions are shared across scenarios; operational decisions and costs are scenario-specific:

\[ \min \; \text{Annuity} + \sum_{\omega\in\Omega} p_\omega \Big( \text{FixedO}\&\text{M}_{\omega} + \text{FuelCost}_{\omega} + \text{GridImportCost}_{\omega} - \text{GridExportRevenue}_{\omega} - \text{Subsidy}_{\omega} + \text{LLCost}_{\omega} + \text{EmissionCost}_{\omega} \Big) \]

Discounting and sizing

In contrast to the multi-year formulation, intertemporal discounting does not affect system sizing in the typical-year model. Costs are evaluated on an annual basis and the system is assumed to operate indefinitely under stationary conditions; annuity-based capital recovery already embeds discounting at the asset level through the WACC. Sizing is therefore driven exclusively by the trade-off between annualized investment costs and expected annual operating costs.

Relationship between the two objectives¶

Both objectives rely on the same bottom-up cost accounting. The typical-year EAC is the steady-state limit of the multi-year objective under time-invariant conditions: the multi-year model resolves investment, operating, and externality costs year by year and discounts them, while the typical-year model annualizes investment over the technical lifetime to yield horizon-independent annual equivalents.