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System Constraints

The system-level constraints couple all technologies and ensure feasibility of supply, reliability, and policy or resource limits. Constraints are enforced at hourly resolution for each scenario; aggregate constraints may additionally apply annually.

Energy balance constraint

At each time step, total net supply equals demand. MicroGridsPy adopts the convention that storage charging and grid export act as demand-side sinks, while storage discharging and grid import act as supply-side sources. Grid flows enter after multiplication by the grid efficiency \(\eta^{\text{grid}}\) (see the PCC convention).

Typical-year formulation

For each \(t\in\mathcal{T}\) and \(\omega\in\Omega\):

\[ \sum_{r} E^{\text{res}}_{t,\omega,r} + E^{\text{gen}}_{t,\omega} + \eta^{\text{grid}} E^{\text{imp}}_{t,\omega} - \eta^{\text{grid}} E^{\text{exp}}_{t,\omega} + E^{\text{dis}}_{t,\omega} - E^{\text{ch}}_{t,\omega} + E^{\text{LL}}_{t,\omega} = D_{t,\omega} \]

where \(\sum_r E^{\text{res}}\) is total renewable generation, \(E^{\text{gen}}\) generator production, \(E^{\text{imp}}/E^{\text{exp}}\) the raw grid flows at the PCC, \(E^{\text{dis}}/ E^{\text{ch}}\) battery discharge/charge on the AC-side balance, \(E^{\text{LL}}\) lost load (a slack supply term ensuring feasibility), and \(D\) demand.

Multi-year formulation

Generator and battery operations are cohort-indexed and summed across investment steps:

\[ \sum_{r} E^{\text{res}}_{t,y,\omega,r} + \sum_k E^{\text{gen}}_{t,y,\omega,k} + \eta^{\text{grid}} E^{\text{imp}}_{t,y,\omega} - \eta^{\text{grid}} E^{\text{exp}}_{t,y,\omega} + \sum_k E^{\text{dis}}_{t,y,\omega,k} - \sum_k E^{\text{ch}}_{t,y,\omega,k} + E^{\text{LL}}_{t,y,\omega} = D_{t,y,\omega} \]

Minimum renewable penetration

A minimum renewable-penetration constraint can represent policy or sustainability targets. It is defined on served supply components, not total load: renewable generation contributes to both the numerator and the denominator; generator output contributes only to the denominator; only the renewable share \(\rho^{\text{grid}}\in[0,1]\) of delivered grid imports contributes to the numerator; lost load and exports do not enter the ratio.

Typical-year formulation

For each scenario \(\omega\):

[ E^{\text{tot}}{\omega} = \sum_t \left( \sum_r E^{\text{res}}} + E^{\text{gen}{t,\omega} + \eta^{\text{grid}} E^{\text{imp}} \right), ] [ E^{\text{ren}}{\omega} = \sum_t \left( \sum_r E^{\text{res}} \right) ]} + \rho^{\text{grid}}\,\eta^{\text{grid}} E^{\text{imp}}_{t,\omega

The constraint is imposed either scenario-wise or in expectation:

\[ E^{\text{ren}}_{\omega} \ge \alpha^{\text{ren}}_{\min,\omega}\, E^{\text{tot}}_{\omega} \quad \forall\omega, \qquad\text{or}\qquad \sum_{\omega} p_{\omega} E^{\text{ren}}_{\omega} \ge \alpha^{\text{ren}}_{\min} \sum_{\omega} p_{\omega} E^{\text{tot}}_{\omega} \]

Multi-year formulation

The same logic applies year by year, with \(E^{\text{tot}}_{y,\omega}\) and \(E^{\text{ren}}_{y,\omega}\) defined analogously (summing generator output over cohorts):

\[ E^{\text{ren}}_{y,\omega} \ge \alpha^{\text{ren}}_{\min,y,\omega}\, E^{\text{tot}}_{y,\omega} \quad \forall y,\omega, \qquad\text{or}\qquad \sum_{\omega} p_{\omega} E^{\text{ren}}_{y,\omega} \ge \alpha^{\text{ren}}_{\min,y} \sum_{\omega} p_{\omega} E^{\text{tot}}_{y,\omega} \quad \forall y \]

Maximum lost-load share

Reliability can be enforced through an upper bound on the fraction of demand that may remain unserved. Typical-year, with \(E^{\text{LL}}_{\omega} = \sum_t E^{\text{LL}}_{t,\omega}\) and \(E^{\text{dem}}_{\omega} = \sum_t D_{t,\omega}\):

\[ E^{\text{LL}}_{\omega} \le \alpha^{\text{LL}}_{\max,\omega}\, E^{\text{dem}}_{\omega} \quad \forall\omega, \qquad\text{or}\qquad \sum_{\omega} p_{\omega} E^{\text{LL}}_{\omega} \le \alpha^{\text{LL}}_{\max} \sum_{\omega} p_{\omega} E^{\text{dem}}_{\omega} \]

Multi-year, enforced year by year:

\[ E^{\text{LL}}_{y,\omega} \le \alpha^{\text{LL}}_{\max,y,\omega}\, E^{\text{dem}}_{y,\omega} \quad \forall y,\omega, \qquad\text{or}\qquad \sum_{\omega} p_{\omega} E^{\text{LL}}_{y,\omega} \le \alpha^{\text{LL}}_{\max,y} \sum_{\omega} p_{\omega} E^{\text{dem}}_{y,\omega} \quad \forall y \]

Constraint enforcement mode

For both minimum renewable penetration and maximum lost-load share, MicroGridsPy supports two enforcement philosophies: scenario-wise, where the constraint holds separately for each scenario, and expected, where it holds only in probability-weighted expectation across scenarios. In the typical-year formulation the aggregates are computed over the representative year; in the multi-year formulation, separately for each modelled year.

Spatial limits on renewable deployment are handled by the land-availability constraint. Together with the economic objective, these constraints let technical feasibility, reliability, and policy requirements be represented explicitly.