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    Dispatch Optimisation

    This section describes the formulation of the dispatch optimisation model in MUSE2. For a given milestone year (MSY), the dispatch model calculates the least-cost operation of the energy system assets to satisfy commodity demands. It is solved as a Linear Programme (LP) using the HiGHS solver.

    Sets and Indices

    The energy system is defined using the following sets:

    • \( r \in \mathbf{R} \): Regions. Represents distinct geographical areas.
    • \( t \in \mathbf{T} \): Time Slices. Discrete operational sub-periods within a year.
    • \( s \in \mathbf{S} \): Time Slice Selections. Collections of time slices representing temporal groupings at different levels (e.g. a single time slice, a season, or the entire year).
    • \( a \in \mathbf{A} \): Assets. All commissioned production, consumption, or conversion technologies.
    • \( c \in \mathbf{C} \): Commodities. All energy carriers, emissions, or materials, partitioned into:
      • \( \mathbf{C}^{\mathrm{SVD}} \): Service Demand commodities (representing final end-use demands).
      • \( \mathbf{C}^{\mathrm{SED}} \): Supply-Equals-Demand commodities (intermediate carrier flows like electricity or hydrogen).
      • \( \mathbf{C}^{\mathrm{OTH}} \): Other commodities (e.g. emissions or tracked flows not subject to supply-demand balancing).

    Decision Variables

    The dispatch model determines the activity level of all assets \( a \in \mathbf{A} \) in all time slices \( t \in \mathbf{T}\):

    \[ \mathrm{Activity}_{a, t} \ge 0 \text{ for all } a \in \mathbf{A}, t \in \mathbf{T} \]

    Activity is a dimensionless quantity representing the level of operation of the asset. It is multiplied by flow coefficients to give the corresponding commodity flows in the units of that commodity (e.g. PJ of energy for an energy flow).

    Objective Function

    The objective is to minimise the total operating cost of the energy system:

    \[ \text{Minimise } \sum_{a \in \mathbf{A}} \sum_{t \in \mathbf{T}} \mathrm{Activity}_{a, t} \cdot \mathrm{Cost}_{\mathrm{Activity},a,t} \]

    Activity Cost Coefficient \( \mathrm{Cost}_{\mathrm{Activity},a,t} \)

    The cost per unit of activity for asset \( a \) in time slice \( t \) is composed of variable operating costs, flow costs, and levies:

    \[ \mathrm{Cost}_{\mathrm{Activity},a,t} = \mathrm{VariableOpex}_a + \sum_{f \in \mathrm{Flows}_a} |f_{\mathrm{coeff}}| \cdot \left( f_{\mathrm{cost}} + \mathrm{Levy}_{f,t} \right) \]

    • \( \mathrm{VariableOpex} \): The non-fuel variable O&M cost of the process.
    • \( f_{\mathrm{coeff}} \): The flow coefficient for a commodity flow associated with the asset (positive for output/production, negative for input/consumption).
    • \( f_{\mathrm{cost}} \): The cost per unit of commodity flow.
    • \( \mathrm{Levy}_{f,t} \): The regional levy (or subsidy, if negative) per unit of commodity flow in time slice \( t \).

    Constraints

    Asset Activity Limits

    For each asset \( a \) and every time slice selection \( s \):

    \[ \mathrm{Avail}_{\mathrm{LB},a,s} \cdot \Delta_s \cdot \mathrm{Capacity}_a \cdot \mathrm{cap2act}_a \le \sum_{t \in s} \mathrm{Activity}_{a, t} \le \mathrm{Avail}_{\mathrm{UB},a,s} \cdot \Delta_s \cdot \mathrm{Capacity}_a \cdot \mathrm{cap2act}_a \]

    where:

    • \( \mathrm{Capacity}_a \) is the fixed installed capacity of the asset.
    • \( \mathrm{cap2act}_a \) is the conversion factor from capacity to activity units.
    • \( \Delta_s = \sum_{t \in s} \Delta_t \) is the total duration of selection \( s \) as a fraction of the year.
    • \( \mathrm{Avail}_{\mathrm{LB},a,s} \) and \( \mathrm{Avail}_{\mathrm{UB},a,s} \) are the lower and upper availability fractions from process_activity_limits.csv, defaulting to \( 0 \) and \( 1 \) respectively for any selection not explicitly defined.

    Equal Utilisation of Equivalent Assets

    To avoid arbitrarily utilising one asset over another, the dispatch model adds additional constraints to equalise the utilisation of assets with equivalent dispatch properties. Assets are considered equivalent when they are in the same region and have the same variable operating cost, activity limits, and commodity flows. Their capacity, state, and process identity are not considered when determining equivalence.

    For an asset \(a\) in time slice \(t\), utilisation is defined as

    \[ \mathrm{Utilisation}_{a,t} = \frac{\mathrm{Activity}_{a,t}}{\mathrm{Capacity}_a \cdot \mathrm{cap2act}_a} \]

    For every pair of equivalent assets \( x \) and \( y \), and for every time slice \( t \), the optimisation model imposes:

    \[ \mathrm{Utilisation}_{x,t} = \mathrm{Utilisation}_{y,t} \]

    Commodity Balance Constraints

    For each balanced commodity \( c \in \mathbf{C}^{\mathrm{SED}} \cup \mathbf{C}^{\mathrm{SVD}} \) in region \( r \) and time slice selection \( s \) at the commodity’s temporal resolution (time_slice_level), the sum of production across all assets minus consumption must satisfy demands:

    \[ \sum_{a \in \mathbf{A}_r} f_{\mathrm{coeff},a,c} \cdot \sum_{t \in s} \mathrm{Activity}_{a, t} \ge \mathrm{Bound}_{c, r, s} \]

    where:

    • \( f_{\mathrm{coeff},a,c} \) is the flow coefficient of commodity \( c \) for asset \( a \) (positive for outputs, negative for inputs).
    • \( \mathrm{Bound}_{c, r, s} \) is the constraint lower bound:
      • For Service Demand (SVD): \( \mathrm{Demand}_{c, r, s} \)
      • For Supply-Equals-Demand (SED): \( 0 \)

    Commodity Consumption/Production Constraints

    Commodity constraints impose lower and upper limits on the total production or consumption of a commodity in a region over a specified time slice selection:

    \[ L_{c,r,s} \leq \sum_{a \in \mathbf{A}_r^d} |f_{\mathrm{coeff},a,c}| \cdot \sum_{t \in s} \mathrm{Activity}_{a,t} \leq U_{c,r,s} \]

    where:

    • \( d \) is the balance type: production (prod) or consumption (cons).
    • \( \mathbf{A}_r^d \) contains assets in region \( r \) with flows in direction \( d \).
    • \( L_{c,r,s} \) and \( U_{c,r,s} \) are the lower and upper limits.

    These constraints are defined in the optional commodity_constraints.csv file. They can apply to SED and OTH commodities, but not SVD commodities. The feature is experimental and requires please_give_me_broken_results = true in model.toml.

    Shadow Prices

    The dual values (shadow prices) of the commodity balance constraints represent the marginal cost of satisfying an additional unit of demand for that commodity in region \( r \) during selection \( s \). These shadow prices are critical outputs of the dispatch model and are used to seed and guide investment appraisal in subsequent steps.

    Seasonal/Annual Utilisation Penalties

    MUSE2 optionally applies small penalties to the peak capacity required by each asset within a season and across the whole year. These penalties encourage activity to be distributed across time slices within a season and across seasons, respectively. They are particularly useful when commodities are balanced at the seasonal/annual levels and the balance constraint otherwise leaves the intra-seasonal or inter-seasonal production profile undetermined. In real-world terms, this represents a preference to avoid concentrating an asset’s operation into short periods of high utilisation, which may reduce cycling, wear, start-up requirements, or the need to maintain capacity for seasonal peaks.

    For asset \(a\) and time slice \(t\), the capacity required to support its activity is

    \[ \mathrm{RequiredCapacity}_{a,t} = \frac{\mathrm{Activity}_{a,t}} {\mathrm{cap2act}_a \cdot \Delta_t} \]

    For each asset and season, MUSE2 introduces an auxiliary variable \(U_{a,s}\), representing the peak capacity required by the asset during that season. It is constrained by

    \[ U_{a,s} \geq \mathrm{RequiredCapacity}_{a,t} \]

    for every time slice \(t\) in season \(s\). In addition, MUSE2 introduces an auxiliary variable \(U_{a,\mathrm{annual}}\), representing the greatest seasonal peak capacity required by the asset during the year. It is constrained by

    \[ U_{a,\mathrm{annual}} \geq U_{a,s} \]

    for every season \(s\).

    When enabled, the penalties add the following term to the optimisation objective:

    \[ \lambda_{\mathrm{seasonal}} \sum_{a \in \mathbf{A}} \sum_{s \in \mathbf{S}} \Delta_s U_{a,s} + \lambda_{\mathrm{annual}} \sum_{a \in \mathbf{A}} U_{a,\mathrm{annual}} \]

    Here, \(\lambda_{\mathrm{seasonal}}\) and \(\lambda_{\mathrm{annual}}\) are set by the seasonal_utilisation_penalty and annual_utilisation_penalty model parameters, respectively. Setting either parameter to zero disables its corresponding penalty. The seasonal parameter controls how strongly activity is spread within seasons, while the annual parameter controls how strongly it is spread across seasons. Both are weighted objective terms, so their values should be small enough that smoothing dispatch does not outweigh meaningful differences in operating cost (default for seasonal_utilisation_penalty and annual_utilisation_penalty are 1e-6 and 1e-5 respectively).

    Candidate Dispatch Run

    After the primary dispatch run, MUSE2 performs a second dispatch run that includes candidate assets — technologies not yet commissioned but available for investment — alongside the existing assets. The purpose of this run is to obtain shadow prices for commodities that are not consumed or produced in the primary dispatch solution.

    In the primary dispatch, a commodity balance constraint will only have a non-zero shadow price if the corresponding constraint is binding (i.e. supply exactly meets demand). For a commodity that is neither produced nor consumed by any active asset, the constraint is trivially satisfied and its dual value is zero. This gives no signal to the investment model about the potential value of new capacity for that commodity.

    By including candidate assets in the second run, MUSE2 ensures that every commodity served by at least one candidate process has a balance constraint that can be binding, yielding a meaningful shadow price. To guarantee a non-zero shadow price even when there is no existing demand for the commodity, a small epsilon is added to the lower bound of commodity balance constraints where candidate assets are present:

    \[ \mathrm{Bound}_{c, r, s} = \begin{cases} \max\left( \mathrm{Demand}_{c, r, s}, \epsilon \right) & \text{if } c \in \mathbf{C}^{\mathrm{SVD}} \text{ and candidate assets serve } (c, r, s) \\ \epsilon & \text{if } c \in \mathbf{C}^{\mathrm{SED}} \text{ and candidate assets serve } (c, r, s) \\ \mathrm{Demand}_{c, r, s} & \text{if } c \in \mathbf{C}^{\mathrm{SVD}} \\ 0 & \text{if } c \in \mathbf{C}^{\mathrm{SED}} \end{cases} \]

    where \( \epsilon \) is the commodity_balance_epsilon parameter. The shadow prices from this candidate dispatch run are then used to seed and guide investment appraisal in subsequent steps.

    Diagnosing Infeasible Models

    In practice, a dispatch optimisation run may be infeasible for several reasons, such as insufficient installed asset capacity to meet demand or incompatible commodity production/consumption constraints. When this occurs, MUSE2 performs additional dispatch runs with modified optimisation problems to help identify the cause. The resulting diagnostic information is included in error messages and saved in the dispatch debug files.

    Unmet Demand Diagnostic

    1. Diagnostic Re-Run: MUSE2 reruns the dispatch optimisation with a set of slack variables representing unmet demand, \( \mathrm{UnmetD}_{c, r, t} \ge 0 \), added to the commodity balance constraints: \[ \sum_{a \in \mathbf{A}_r} f_{\mathrm{coeff},a,c} \cdot \sum_{t \in s} \mathrm{Activity}_{a, t} + \sum_{t \in s} \mathrm{UnmetD}_{c, r, t} \ge \mathrm{Bound}_{c, r, s} \]
    2. Objective Penalty: To ensure the solver only leaves demand unmet if it is mathematically impossible to satisfy it, these variables are heavily penalised in the diagnostic objective function using the value_of_lost_load parameter (\( \mathrm{VoLL} \)): \[ \text{Minimise } \sum_{a \in \mathbf{A}} \sum_{t \in \mathbf{T}} \mathrm{Activity}_{a, t} \cdot \mathrm{Cost}_{\mathrm{Activity},a,t} + \mathrm{VoLL} \cdot \sum_{c, r, t} \mathrm{UnmetD}_{c, r, t} \]
    3. Isolating Shortfalls: The addition of \( \mathrm{UnmetD}_{c, r, t} \) guarantees that the LP remains mathematically feasible. When solved, any time slice, region, or commodity with a shortfall will have \( \mathrm{UnmetD}_{c, r, t} > 0 \).
    4. Error Reporting: MUSE2 scans the solution, identifies all balanced markets \( (c, r) \) where unmet demand occurred, outputs detailed diagnostic CSV files, and aborts the simulation with an error identifying the exact out-of-balance markets.

    Commodity Constraints Diagnostic

    If the dispatch optimisation remains infeasible, MUSE2 reruns it with the commodity consumption/production constraints disabled. If this rerun succeeds, the infeasibility is likely caused by one or more constraints defined in commodity_constraints.csv.

    If the rerun remains infeasible, commodity constraints are not identified as the cause. Since commodity constraints are an experimental feature, this diagnosis should be treated as indicative rather than definitive.