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Energy Balances

The energy balance equations are the most important constraints, which enforces that incoming and outgoing energy flows balance out at each bus \(n\) for each time \(t\) (like Kirchhoff's current law for electrical buses, see Linearised Power Flow). Considering all components, the balance constraint is given by

\[\begin{gather*}\sum_{s} g_{n,s,t} + \sum_{s} \left(h_{n,s,t}^- - h_{n,s,t}^+ \right) + \sum_{s} h_{n,s,t}\\ + \sum_{l} L_{n,l,t} f_{l,t} + \sum_{l} K_{n,l} p_{l,t} = \sum_{s} d_{n,s,t} \quad \leftrightarrow \quad w_t^o\lambda_{n,t}\end{gather*}\]

where the decision variables are represented by:

  • \(d_{n,s,t}\) for the demand of Load components
  • \(g_{n,s,t}\) for the dispatch of Generator components
  • \(h_{n,s,t}^-\) for the discharging of StorageUnit components
  • \(h_{n,s,t}^+\) for the charging of StorageUnit components
  • \(f_{l,t}\) for the flow on the link Link at bus0
  • \(p_{l,t}\) for the power flow on the Line and Transformer components at bus0

Note

The orientation of each dispatch variable can be inverted or rescaled through the sign attribute of the component.

The incidence matrices \(K_{n,l}\) for the Line and Transformer components and \(L_{n,l,t}\) for the Link components govern flows between buses. The incidence matrix \(K_{n,l}\) takes non-zero values \(-1\) if the line or transformer \(l\) starts at bus \(n\) and \(1\) if it ends at bus \(n\). If \(p_{l,t}>0\) it withdraws from the starting bus. The time-varying incidence matrix \(L_{n,l,t}\) takes non-zero values \(-1\) if the link \(l\) starts at bus \(n\) and efficiency \(\eta_{n,l,t}\) if it ends at bus \(n\). If \(f_{l,t}>0\) it withdraws from bus0 and feeds in \(\eta_{n,l,t} f_{l,t}\) to bus1. For a link with more than two outputs (e.g. a combined heat and power plant), the incidence matrix \(L_{n,l,t}\) has more than two non-zero entries with efficiencies \(\eta_{n,l,t}\) for bus2, bus3, etc.. The entries may also be negative to denote additional inputs rather than multiple outputs.

The dual variable \(\lambda_{n,t}\) represents the shadow price of the constraint (e.g. market clearing price, dynamic locational marginal prices) and is scaled by the snapshot weighting \(w_t^o\) to yield units of currency per unit of energy regardless of the time resolution.

The energy balance constraints are set in the function define_nodal_balance_constraints() and is called Bus-nodal_balance.

Mapping of symbols to component attributes
Symbol Attribute Type
\(g_{n,s,t}\) n.generators_t.p Decision variable
\(h_{n,s,t}^-\) n.storage_units_t.p_dispatch Decision variable
\(h_{n,s,t}^+\) n.storage_units_t.p_store Decision variable
\(h_{n,s,t}\) n.stores_t.p Decision variable
\(f_{l,t}\) n.links_t.p0 Decision variable
\(p_{l,t}\) n.lines_t.p0 or n.transformers_t.p0 Decision variable
\(d_{n,s,t}\) n.loads_t.p_set Decision variable
\(\lambda_{n,t}\) n.buses_t.marginal_price Dual variable
\(K_{n,l}\) Calculated internally by n.incidence_matrix() Parameter
\(L_{n,l,t}\) Calculated internally from efficiency{i} attributes Parameter
\(w_t^o\) n.snapshot_weightings.objective Parameter