Backup Generators¶

MicroGridsPy models dispatchable backup generation through a single generator technology representing the backup unit of the mini-grid, described by a unit-based sizing variable and an hourly production variable. Fuel consumption is modelled explicitly and linked to electrical output through either a nominal-efficiency relationship or a partial-load model that combines an affine Willans fuel line with a clustered unit-commitment variable.

The same structure is used in both modes; in the multi-year formulation, generator investment and operation are cohort-based, so production limits and fuel relations are indexed by year \(y\) and step \(k\). At hourly resolution \(\Delta t = 1\,\text{h}\), generation in kWh over one step equals average power in kW.

Installed capacity and production limit¶

Typical-year formulation¶

The generator is sized in discrete units \(N^{\text{gen}}\), each with nominal capacity \(P^{\text{gen}}\) (kW), for a total \(C^{\text{gen}} = N^{\text{gen}} \cdot P^{\text{gen}}\). Hourly production is bounded by installed capacity, with an optional maximum-installable bound:

\[ E^{\text{gen}}_{t,\omega} \le N^{\text{gen}} \cdot P^{\text{gen}} \qquad \forall t,\omega, \qquad\qquad N^{\text{gen}} \cdot P^{\text{gen}} \le \overline{C}^{\text{gen}} \]

Multi-year formulation¶

Generator investment is cohort-based. Each step \(k\) introduces \(N^{\text{gen}}_{k}\) units, with nominal cohort capacity \(C^{\text{gen}}_{k} = N^{\text{gen}}_{k} \cdot P^{\text{gen}}\). The available cohort capacity in year \(y\) is

\[ \widetilde{C}^{\text{gen}}_{y,k} = N^{\text{gen}}_{k}\cdot P^{\text{gen}}\cdot \alpha_{y,k}\cdot \delta_{y,k} \]

where \(\alpha_{y,k}\) is the cohort activity/replacement mask and \(\delta_{y,k}\) an optional exogenous degradation factor. Production is bounded cohort by cohort and aggregated:

\[ E^{\text{gen}}_{t,y,\omega,k} \le \widetilde{C}^{\text{gen}}_{y,k} \qquad \forall t,y,\omega,k, \qquad\qquad E^{\text{gen,tot}}_{t,y,\omega} = \sum_k E^{\text{gen}}_{t,y,\omega,k} \]

The maximum-installable bound applies to the cumulative design: \(\sum_k N^{\text{gen}}_{k} \cdot P^{\text{gen}} \le \overline{C}^{\text{gen}}\).

Fuel–power relationship (nominal efficiency)¶

When partial-load modelling is disabled, fuel consumption is linked to output through a constant nominal efficiency \(\eta^{\text{nom}}\) and the fuel lower heating value \(\text{LHV}\):

\[ E^{\text{gen}}_{t,\omega} = F_{t,\omega}\cdot \text{LHV}\cdot \eta^{\text{nom}} \qquad\text{(typical-year)} \]

\[ E^{\text{gen}}_{t,y,\omega,k} = F_{t,y,\omega,k}\cdot \text{LHV}\cdot \eta^{\text{nom}} \qquad\text{(multi-year)} \]

where \(F\) is fuel consumption in units consistent with the LHV. The implied specific fuel consumption is constant over the whole operating range.

Partial-load efficiency and unit commitment¶

When partial-load modelling is enabled, efficiency becomes output-dependent. A real diesel genset burns fuel just to stay running (a no-load intercept) plus extra fuel per unit of output, so specific fuel consumption worsens sharply at low load. This is captured by an affine Willans fuel line paired with a committed-capacity variable.

Willans fuel line¶

The user-provided efficiency curve gives relative loading points \(r_b \in (0,1]\) with efficiencies \(\eta_b\). The implied relative fuel-use is \(\phi(r) = r/\eta(r)\), which is fit to the affine function

\[ \phi(r) = q_0 + q_1\, r \]

anchored at full load so the datasheet full-load efficiency is preserved exactly (\(q_0 + q_1 = 1/\eta^{\text{nom}}\)). Here \(q_0 \ge 0\) is the relative no-load fuel use and \(q_1 > 0\) the marginal relative fuel use. Because the origin is handled by the commitment variable below, the curve is not anchored at \((0,0)\) and no convex majorant is needed.

Committed capacity and unit commitment¶

A commitment variable \(N^{\text{on}}\) counts the generator units online, so the online capacity is \(N^{\text{on}} P^{\text{gen}}\). Output is bounded by the online capacity (with an optional minimum stable load \(m \in [0,1)\)), the online count cannot exceed the available capacity, and fuel carries the no-load intercept per online unit:

\[ \begin{aligned} m\, N^{\text{on}}_{t,\omega} P^{\text{gen}} \;\le\; E^{\text{gen}}_{t,\omega} &\;\le\; N^{\text{on}}_{t,\omega} P^{\text{gen}} \\[3pt] N^{\text{on}}_{t,\omega} P^{\text{gen}} &\;\le\; N^{\text{gen}} P^{\text{gen}} \\[3pt] F_{t,\omega} &\;\ge\; \frac{q_1}{\text{LHV}}\, E^{\text{gen}}_{t,\omega} + \frac{q_0}{\text{LHV}}\, N^{\text{on}}_{t,\omega} P^{\text{gen}} \end{aligned} \qquad \forall t,\omega \]

Minimising fuel makes the epigraph tight, so idling committed capacity burns the no-load fuel even at zero output. In the multi-year formulation the same relations are written per cohort \(k\), with the online capacity \(N^{\text{on}}_{t,y,\omega,k} P^{\text{gen}}\) bounded by the available (degraded) cohort capacity \(\widetilde{C}^{\text{gen}}_{y,k}\).

Commitment modes¶

Mode	\(N^{\text{on}}\)	Behaviour
`off`	—	constant full-load efficiency (nominal relationship)
`integer`	integer	whole online units (clustered unit commitment, after Palmintier & Webster). The no-load fuel and the minimum stable load become binding, so the genset refuses sub-minimum loads and pays the part-load penalty

Clustered integer commitment uses a single integer per timestep (per cohort), keeping the mixed-integer problem far smaller than one binary per unit while still capturing on/off and minimum-load physics.

Generator efficiency: left, a real efficiency curve compared with a constant-efficiency approximation; right, the curve sampled at relative-output points used to fit the Willans fuel line

Generator efficiency under the constant and partial-load formulations. Left: a real generator efficiency curve versus the constant-efficiency approximation — real efficiency falls sharply at low load because of the no-load fuel intercept. Right: the sampled efficiency points used to fit the affine Willans fuel line, which preserves the datasheet full-load efficiency.

What is and is not modelled

The Willans fit preserves the datasheet full-load efficiency and introduces a genuine no-load fuel intercept; the input curve is validated at model build (strictly positive efficiencies, a non-decreasing implied fuel curve). The integer mode adds a minimum stable load and true on/off behaviour, but start-up and shut-down costs and minimum up/down times are not yet modelled. Part-load commitment is a mixed-integer program; over long multi-year horizons pair it with representative periods to keep it tractable.