Model Formulation¶
The market model follows the typical structure of an econonomic dispatch, where the generation schedule of each power plant is determined based on the short run marginal costs and the availability of renewable energy sources (RES). Therefore, the model represents an optimization problem minimizing total generation costs subject to constraints regarding generation capacity and transport limitations. Which constraints are present depends on the options.
Each power plant is categorized by its plant type. Depending on plant type different subsets of constraints apply to its generation variables. The model distinguishes between the following plant types:
\(\mathcal{P}\): Conventional power plants.
\(\mathcal{TS}\): Plants that have a timedependant capacity factor (availability).
\(\mathcal{ES} \subset \mathcal{P}\) : Electricity storages.
\(\mathcal{HE} \subset \mathcal{P}\): Plants that produce heat.
\(\mathcal{CHP} \subset \mathcal{P}\): Plants that provide both heat and electricity.
\(\mathcal{PH} \subset \mathcal{P}\): Power to heat plants.
\(\mathcal{HS} \subset \mathcal{P}\): Heat Storages.
The generation of a power plant can be characterized by different decision variables, depending on plant type, which are as follows:
\(G\): Electricity generation
\(D\): Electricity demand, for plants of type ES and PH.
\(\mathit{CURT}\): Curtailment variable for plant of type TS.
\(L^{es}\): Storage level for electricity storages (plant type ES).
\(H\): Heat generation for all plants of type HE.
\(L^{hs}\): Storage level for heat storages (plant type HS).
Generation Constraints¶
Electricity generation is bound by the installed capacity per plant. For RES the generation is determined by the availability minus the curtailment variable.
Storages are represented by a storage level, that redices by generation and increases by charging/demand. Storage efficiency reduces the electricity stored.
Electricity generation occurs costs for fuel and possibly curtailment, which will be included in the objective function:
Energy Balance¶
Due to transport constraints applying to both the zonal energy balance, i.e. the net position (NEX) represented by the exchange variable EX which represents commercial exchange between two market areas, the nodal balance, both balances have to be explicitly made and utilize mapping between the power plant sets and their nodal/zonal affiliation. Generally, the zonal energy balance allows to constrain commercial exchange while the nodal energy balance set the nodal injection (INJ) and therefore is constrained by the transmission network or its representation.
Transport Constraints¶
Based on the chosen configuration, tranport is constrained either as commercial exchange, i.e. in between market areas, or based on the nodal injections in a more technically accurate representation on the transmission system.
Physical flows are modeled using some modification of the linearized (DC) power flow equations so that nodal net injections can be mapped to power flows via a power transfer distribution factor (PTDF) matrix or voltage angles. Both formulations come with certain advantages and disadvantages.
The formulations include the line parameters susceptance \(x\), PTDF matrices, line capacities \(f^{max}\), incidence matrix \(A\) which are calculated beforehand and available to the julia model core.
Depending in the chosen configuration different variables, sets or constraints are activated.
Angle Formulation¶
Power flow can be implemenetad via the nodal voltage angles \(\Theta_{n,t}\). The benifit is, compared the PTDF formulation a significantly lower memory usage.
The disadvante is that a \(\Theta\) has to be set for each node in a connected network, regardless of how many lines are to be considered with thermal capacities. Considering only a subset of lines in the dispatch does therefore not pose significant advantages. This disadvantage becomes more impactful when including contingencies, which are accommodated as contingency scenarios \(\mathcal{C}\) (which include potentially multiple outages) for \(\Theta\) and the flow \(F\), effectively enforcing constraints on multiple flow scenarios for the same net-injections. The incidence matrix \(A^c\) includes the topology changes caused by the contingency scenario and the flow on outed lines is forced to zero.
This formulation would be efficient if we would like to concider all lines for each contingency. This is however never the case and the utilization of the RedundandyRemoval has shown that less then 1% of all contingencies are needed to ensure SCOPF.
In addition, zonal PTDF represent a line specific zonal aggregation of networks, which cannot be directly implemented in an angle formulation.
PTDF Formulation¶
The PTDF formulation of linear power flow uses power transfer distribution factors (PTDF) to map nodal injections to line flows. These are calculated beforehand using the network topology, line parameters and contingencies.
Note that the DCLF constraints are intentionally written in matrix form instead of elementwise like above. Thereby the formulation is more general, which reflects the actual formulation in which the PTDF can accommodate any power flow configuration POMATO offers. This can be N-0 (nodal pricing, OPF), N-1 (SCOPF) in a reduced representation or full, including combined contingencies (n-k) or specified contingency groups. Assinging the line flows to two positive variables reduces the model complexity, as the dense and possibly extremely large PTDF matrix is only used once per timestep.
The PTDF allows for flexible selection of which lines and outages to consider in the economic dispatch and propotionally increases complexity with the number of considered contingency cases.
An analogue formulation applies for a zonal PTDF, except that the line flows result from the NEX (exports minus imports). Note that the PTDF is denoted with index t, indicating a potentially time dependant PTDF as used in the implementation of Flow Based Market Coupling (FBMC) and the FB Domain.
The zonal PTDF is computed based on the given configuration and relies on weighting parameters that convert the zonal net position NEX into nodal injections. This concept is farmally defined within FBMC as a generation shift key (GSK).
POMATO will choose the angle and PTDF formulation depending on the chosen configuration. Generally, when most lines are part of the optimization, like in nodal pricing, the angle formulation is much faster. When only a small subset of lines is relevant or contingency cases are used, the PTDF formulation is better. Zonal application will always use the PTDF formulation.
Commercial Exchange and Flow on DC-Lines¶
Beside nodal/zonal transmission network representations, tranport constraints can be includes as net trans capacities (NTC), that directly constraint the commercial exchange. These constraints can be configured to only apply to commercial exchange from and to a subset of zones.
DC lines are also constrained to upper and lower bounds. DC lines are modeled as part of the market result and their power flow is optimized with system cost in mind.
The flow on a dc line is mapped to the start and endnodes using the \(\mathrm{incidence}_{dc,n}\) parameter and is included in the nodal energy balance.
Both commercial exchange and flows on dc lines are decision variables. This can cause unintended behavior, where the absolute values of flows can be high when only condition is that all flows are balanced. Therefore there are small costs associated with both commercial flows and flows on dc lines that are captured as \(\mathit{COST\_EX}\).
Heat-Generation Constraints¶
The model can accommodate generation of heat into the economic dispatch problem. However, the additional data needed is difficult to come by. The concept is, that heatareas \(\mathcal{HA}\) are defined analog to market areas and a heat demand for each heatarea has to be balanced by plants which are located within. Plants are subject to a maximum generation and co-generation of heat and electricity is constraints by additional constraints. There can be heat generated by plants of type \(\mathcal{TS}\), but it cannot be curtailed.
The generation from CHP is modeled with 2-degrees of freedom, where the first constraint represents the extraction line, and the second constraint the upper-bound for heat and electricity generation. Generally, CHP can be modeled with much greater detail, however the heat formulation’s purpose is to allow to roughly model the adjacent sector and allow for soft must-run constraints.
Plants of type \(\mathcal{PH}\) convert an electricity demand into heat and heat storages \(\mathcal{HS}\) can shift heat generation to later periods. Note that the inclusion of storages will always greatly increase model complexity.
Heat generation and demand have to be balanced and heat generation will occur costs.
Objective Value¶
The objective value represents the total system cost and consist of all individual cost components and is subject to all constraints layed out above. Note that not all constraints have to be present each model run, but depend on the individual configuration through the options of each run.
