# Dispatch Optimisation Formulation
This dispatch optimisation model calculates the least-cost operation of the energy system for a
given configuration of assets and capacities, subject to demands and constraints. It is the core
engine used for each dispatch run referenced in the overall MUSE2 workflow. A key general
assumption is that SVD commodities represent final demands only and are not consumed as inputs by
any asset.
## General Sets
These define the fundamental categories used to define the energy system.
- \\( \mathbf{R} \\): Set of Regions (indexed by \\( r \\)). Represents distinct geographical or
modelling areas.
- \\( \mathbf{T} \\): Set of Time Slices (indexed by \\( t \\)). Discrete operational periods within
a year.
- \\( \mathbf{H} \\): Set of Seasons (indexed by \\( h \\)). Collections of time slices.
- \\( \mathbf{A} \\): Set of All Assets (indexed by \\( a \\)). All existing and candidate
production, consumption, or conversion technologies.
- \\( \mathbf{A}^{flex} \subseteq \mathbf{A} \\): Subset of Flexible Assets (variable input/output
ratios).
- \\( \mathbf{A}^{std} = \mathbf{A} \setminus \mathbf{A}^{flex} \\): Subset of Standard Assets
(fixed input/output coefficients).
- \\( \mathbf{C} \\): Set of Commodities (indexed by \\( c \\)). All energy carriers, materials, or
tracked flows. Partitioned into:
- \\( \mathbf{C}^{\mathrm{SVD}} \\): Supply-Driven Commodities (final demands; not consumed by
assets).
- \\( \mathbf{C}^{\mathrm{SED}} \\): Supply-Equals-Demand Commodities (intermediate system flows
like grid electricity).
- \\( \mathbf{C}^{\mathrm{OTH}} \\): Other Tracked Flows (e.g., losses, raw emissions).
- \\( \mathbf{C}^{VoLL} \subseteq \mathbf{C}^{\mathrm{SVD}} \cup \mathbf{C}^{\mathrm{SED}} \\):
Subset of commodities where unserved demand is modelled with a penalty.
- \\( \mathbf{P} \\): Set of External Pools/Markets (indexed by \\( p \\)).
- \\( \mathbf{S} \\): Set of Scopes (indexed by \\( s \\)). Sets of \\( (r,t) \\) pairs for policy
application.
## A. Core Model (Standard Assets: \\( a \in \mathbf{A}^{std} \\))
**Purpose:** Defines the operation of assets with fixed, predefined input-output relationships.
### A.1. Parameters (for \\( a \in \mathbf{A}^{std} \\) or global)
- \\( duration[t] \\): Duration of time slice \\( t \\) as a fraction of the year (\\( \in (0,1]
\\)). Represents the portion of the year covered by this slice.
- \\( season\\_ slice[h,t] \\): Binary indicator; \\( 1 \\) if time slice \\( t \\) is in season \\(
h \\), \\( 0 \\) otherwise. Facilitates seasonal aggregation.
- \\( balance\\_ level[c,r] \\): Defines the temporal resolution ('timeslice', 'seasonal', 'annual')
at which the supply-demand balance for commodity \\( c \\) in region \\( r \\) must be enforced.
- \\( demand[r,c] \\): Total annual exogenously specified demand (\\( \ge 0 \\)) for commodity \\( c
\in \mathbf{C}^{\mathrm{SVD}} \\) in region \\( r \\). This is the final demand to be met.
- \\( timeslice\\_ share[c,t] \\): Fraction (\\( \in [0,1] \\)) of the annual \\( demand[r,c] \\)
for \\( c \in \mathbf{C}^{\mathrm{SVD}} \\) that occurs during time slice \\( t \\). (\\(
\sum_{t}timeslice\\_ share[c,t]=1 \\)). Defines the demand profile.
- \\( capacity[a,r] \\): Installed operational capacity (\\( \ge 0 \\)) of asset \\( a \\) in region
\\( r \\) (e.g., MW for power plants). This value is an input to each dispatch run.
- \\( cap2act[a] \\): Conversion factor (\\( >0 \\)) from asset capacity units to activity units,
ensuring consistency between capacity (e.g., MW) and activity (e.g., MWh produced in a slice)
considering \\( duration[t] \\).
- \\( avail_{UB}[a,r,t], avail_{LB}[a,r,t], avail_{EQ}[a,r,t] \\): Availability factors (\\( \in
[0,1] \\)) for asset \\( a \\) in time slice \\( t \\). \\( UB \\) is maximum availability, \\( LB
\\) is minimum operational level, \\( EQ \\) specifies exact operation if required.
- \\( cost_{var}[a,r,t] \\): Variable operating cost (\\( \ge 0 \\)) per unit of activity for asset
\\( a \\) (e.g., non-fuel O&M).
- \\( input_{coeff}[a,c] \\): Units (\\( \ge 0 \\)) of commodity \\( c \in
(\mathbf{C}^{\mathrm{SED}} \cup \mathbf{C}^{\mathrm{OTH}}) \\) consumed by asset \\( a \\) per
unit of its activity. (By assumption, \\( input_{coeff}[a,c]=0 \\) if \\( c \in
\mathbf{C}^{\mathrm{SVD}} \\)).
- \\( output_{coeff}[a,c] \\): Units (\\( \ge 0 \\)) of commodity \\( c \in \mathbf{C} \\) produced
by asset \\( a \\) per unit of its activity.
- \\( cost_{input}[a,c] \\): Specific cost (\\( \ge 0 \\)) per unit of input commodity \\( c \\)
consumed by asset \\( a \\). Useful if \\( c \\) attracts a levy/incentive \\( only \\) if it is
consumed by this type of asset.
- \\( cost_{output}[a,c] \\): Specific cost (if positive) or revenue (if negative) per unit of
output commodity \\( c \\) produced by asset \\( a \\). Useful if levy/incentive applies \\( only
\\) when the commodity is produced by this type of asset.
- \\( VoLL[c,r] \\): Value of Lost Load. A very high penalty cost applied per unit of unserved
demand for \\( c \in \mathbf{C}^{VoLL} \\) in region \\( r \\).
### A.2. Decision Variables
These are the quantities the dispatch optimisation model determines.
- \\( act[a,r,t]\ge0 \\): Activity level of asset \\( a \\) in region \\( r \\) during time slice
\\( t \\). This is the primary operational decision for each asset.
- \\( UnmetD[c,r,t]\ge0 \\): Unserved demand for commodity \\( c \in \mathbf{C}^{VoLL} \\) in region
\\( r \\) during time slice \\( t \\). This variable allows the model to find a solution even if
capacity is insufficient.
### A.3. Objective Contribution (for standard assets \\( a \in \mathbf{A}^{std} \\))
This term represents the sum of operational costs associated with standard assets, forming a
component of the overall system cost that the model seeks to minimise.
\\[
\sum_{a\in \mathbf{A}^{std}}\sum_{r,t} act[a,r,t]
\Biggl(
cost_{var}[a,r,t] +
\sum_{c \notin \mathbf{C}^{\mathrm{SVD}}} cost_{input}[a,c]\\,input_{coeff}[a,c] +
\sum_{c \in \mathbf{C}} cost_{output}[a,c]\\,output_{coeff}[a,c]
\Biggr)
\\]
### A.4. Constraints (Capacity & Availability for standard assets \\( a \in \mathbf{A}^{std} \\))
These constraints ensure that each standard asset's operation respects its physical capacity and
time-varying availability limits. For all \\( a \in \mathbf{A}^{std}, r, t \\):
- Asset activity \\( act[a,r,t] \\) is constrained by its available capacity, considering its
minimum operational level (lower bound, LB) and maximum availability (upper bound, UB):
\\[
\begin{aligned}
capacity[a,r]\\,cap2act[a]\\,avail_{LB}[a,t]\\,duration[t] &\le act[a,r,t] \\\\
act[a,r,t] &\le capacity[a,r]\\,cap2act[a]\\,avail_{UB}[a,t]\\,duration[t]
\end{aligned}
\\]
- If an exact operational level is mandated (e.g., for some renewables based on forecast, or fixed
generation profiles for specific assets):
\\[ act[a,r,t] = capacity[a,r]\\,cap2act[a]\\,avail_{EQ}[a,t]\\,duration[t] \\]
## B. Flexible Assets (\\( a \in \mathbf{A}^{flex} \\))
**Purpose:** This section defines the operational characteristics of flexible assets, which can
adjust their input consumption mix and/or output product shares, governed by an overall process
efficiency. The generic activity variable \\( act[a,r,t] \\) (linked to the asset's physical \\(
capacity[a,r] \\)) is connected to the scale of this flexible conversion process via a reference
output parameter. It is assumed that SVD commodities cannot be efficiency-constrained or auxiliary
inputs to these assets.
### B.1. Asset-Specific Sets (Defined for each flexible asset \\( a \in \mathbf{A}^{flex} \\))
These sets categorize the commodities involved in the flexible asset's operation.
- \\( \mathbf{C}^{eff\\_ in}_a \subseteq (\mathbf{C}^{\mathrm{SED}} \cup \mathbf{C}^{\mathrm{OTH}})
\\): Set of input commodities for asset \\( a \\) whose combined energy or mass content is subject
to the main process efficiency \\( \eta[a] \\).
- \\( \mathbf{C}^{eff\\_ out}_a \subseteq \mathbf{C} \\): Set of main output commodities from asset
\\( a \\) whose combined energy or mass content is determined by \\( \eta[a] \\) and the processed
inputs.
- \\( \mathbf{C}^{aux\\_ in}_a \subseteq (\mathbf{C}^{\mathrm{SED}} \cup \mathbf{C}^{\mathrm{OTH}})
\\): Set of auxiliary input commodities for asset \\( a \\) (e.g., water, catalysts) whose
consumption is typically proportional to the main activity \\( act[a,r,t] \\) but which are not
part of the primary energy/mass balance for the efficiency calculation.
- \\( \mathbf{C}^{aux\\_ out}_a \subseteq \mathbf{C} \\): Set of auxiliary output commodities for
asset \\( a \\) (e.g., specific emissions, waste streams) whose production is typically
proportional to \\( act[a,r,t] \\) but which are not counted in the efficiency calculation for the
primary products.
### B.2. Parameters (for \\( a \in \mathbf{A}^{flex} \\))
These parameters define the technical and economic behavior of flexible assets.
- \\( \eta[a] \\): The overall process efficiency (\\( \in (0,1] \\)) of flexible asset \\( a \\),
relating total common units of outputs in \\( \mathbf{C}^{eff\\_out}_a \\) to inputs in \\(
\mathbf{C}^{eff\\_in}_a \\).
- \\( factor_{CU}[c] \\): A commodity-specific factor to convert its native physical units (e.g.,
tonnes, m³) to a common unit (e.g., MWh of energy content, or tonnes of mass if it's a mass
balance) used for consistent efficiency and input/output share calculations.
- \\( RefEffOutPerAct[a] \\): Reference Total Efficiency-constrained Output per unit of Activity.
This crucial parameter defines the total quantity of efficiency-constrained outputs (summed in
common units using \\( factor_{CU}[c] \\)) that are produced when asset \\( a \\) operates at one
unit of its generic activity level \\( act[a,r,t] \\).
- \\( minInputShare[a,c] \\) (for \\( c \in \mathbf{C}^{eff\\_ in}_a \\)): Minimum fractional share
of input commodity \\( c \\) (in common units) relative to the total efficiency-constrained input
(in common units).
- \\( maxInputShare[a,c] \\) (for \\( c \in \mathbf{C}^{eff\\_ in}_a \\)): Maximum fractional share
for input \\( c \\).
- \\( minOutputShare[a,c] \\) (for \\( c \in \mathbf{C}^{eff\\_ out}_a \\)): Minimum fractional
share of output commodity \\( c \\) (in common units) relative to the total efficiency-constrained
output.
- \\( maxOutputShare[a,c] \\) (for \\( c \in \mathbf{C}^{eff\\_ out}_a \\)): Maximum fractional
share for output \\( c \\).
- \\( coeff_{aux\\_ in}[a,c] \\) (for \\( c \in \mathbf{C}^{aux\\_ in}_a \\)): Quantity of auxiliary
input \\( c \\) consumed per unit of \\( act[a,r,t] \\).
- \\( coeff_{aux\\_ out}[a,c] \\) (for \\( c \in \mathbf{C}^{aux\\_ out}_a \\)): Quantity of
auxiliary output \\( c \\) produced per unit of \\( act[a,r,t] \\).
### B.3. Decision Variables (for \\( a \in \mathbf{A}^{flex} \\))
These variables represent the operational choices for flexible assets.
- \\( act[a,r,t]\ge0 \\): Generic activity level of flexible asset \\( a \\) (linked to its \\(
capacity[a,r] \\)).
- \\( InputSpec[a,c,r,t]\ge0 \quad \forall c \in \mathbf{C}^{eff\\_ in}\_a \\): Actual amount of
efficiency-constrained input commodity \\( c \\) consumed by asset \\( a \\) in region \\( r \\),
time \\( t \\) (in its native physical units).
- \\( OutputSpec[a,c,r,t]\ge0 \quad \forall c \in \mathbf{C}^{eff\\_ out}\_a \\): Actual amount of
efficiency-constrained output commodity \\( c \\) produced by asset \\( a \\) in region \\( r \\),
time \\( t \\) (in its native physical units).
### B.4. Objective Contribution (for \\( a \in \mathbf{A}^{flex} \\))
The cost contribution from flexible assets includes costs related to their generic activity,
specific costs for efficiency-constrained inputs and outputs (if defined separately from system
commodity values), and costs/revenues for auxiliary inputs and outputs.
\\[
\begin{aligned}
&act[a,r,t] \cdot cost\_{var}[a,r,t] \\\\
&+ \sum\_{c \in \mathbf{C}^{eff\\_ in}\_a} InputSpec[a,c,r,t] \cdot cost\_{input}[a,c] \\\\
&+ \sum\_{c \in \mathbf{C}^{eff\\_ out}\_a} OutputSpec[a,c,r,t] \cdot cost\_{output}[a,c] \\\\
&+ \sum\_{c \in \mathbf{C}^{aux\\_ in}\_a} (act[a,r,t] \cdot coeff\_{aux\\_ in}[a,c])
\cdot cost\_{input}[a,c] \\\\
&+ \sum\_{c \in \mathbf{C}^{aux\\_ out}\_a} (act[a,r,t] \cdot coeff\_{aux\\_ out}[a,c])
\cdot cost\_{output}[a,c] \\\\
\end{aligned}
\\]
### B.5. Constraints (for \\( a \in \mathbf{A}^{flex}, r \in \mathbf{R}, t \in \mathbf{T} \\))
These rules govern the internal operation and conversion process of flexible assets. Let \\(
ActualTotalEffOutputCU[a,r,t] = RefEffOutPerAct[a] \cdot act[a,r,t] \\) (This is the total
efficiency-constrained output in common units). Let \\( ActualTotalEffInputCU[a,r,t] =
(RefEffOutPerAct[a] / \eta[a]) \cdot act[a,r,t] \\) (This is the total efficiency-constrained input
in common units, derived from the output and efficiency).
- **Capacity & Availability:** Standard capacity and availability constraints (as in A.4) apply to
the generic activity variable \\( act[a,r,t] \\) of the flexible asset.
- **Total Efficiency-Constrained Input Definition:** The sum of all specific efficiency-constrained
inputs, when converted to common units by \\( factor_{CU}[c] \\), must equal the total
efficiency-constrained input derived from \\( act[a,r,t] \\) and the asset's efficiency:
\\[
\sum\_{c \in \mathbf{C}^{eff\\_ in}\_a} InputSpec[a,c,r,t] \cdot factor\_{CU}[c]
= ActualTotalEffInputCU[a,r,t]
\\]
- **Total Efficiency-Constrained Output Definition:** Similarly, the sum of all specific main
outputs (in common units) must equal the total defined by \\( act[a,r,t] \\) and the reference
output parameter:
\\[
\sum\_{c \in \mathbf{C}^{eff\\_out}\_a} OutputSpec[a,c,r,t] \cdot factor\_{CU}[c]
= ActualTotalEffOutputCU[a,r,t]
\\]
- **Input Share Constraints** (for each \\( c \in \mathbf{C}^{eff\\_in}_a \\)): Ensure that each
individual efficiency-constrained input (in common units) stays within its defined minimum (\\(
minInputShare[a,c] \\)) and maximum (\\( maxInputShare[a,c] \\)) fractional share of the \\(
ActualTotalEffInputCU[a,r,t] \\).
\\[
\begin{aligned}
InputSpec[a,c,r,t] \cdot factor\_{CU}[c]
&\ge minInputShare[a,c] \cdot ActualTotalEffInputCU[a,r,t] \\\\
InputSpec[a,c,r,t] \cdot factor\_{CU}[c]
&\le maxInputShare[a,c] \cdot ActualTotalEffInputCU[a,r,t]
\end{aligned}
\\]
- **Output Share Constraints** (for each \\( c \in \mathbf{C}^{eff\\_out}_a \\)): Ensure that each
individual efficiency-constrained output (in common units) stays within its defined minimum (\\(
minOutputShare[a,c] \\)) and maximum (\\( maxOutputShare[a,c] \\)) fractional share of the \\(
ActualTotalEffOutputCU[a,r,t] \\).
\\[
\begin{aligned}
OutputSpec[a,c,r,t] \cdot factor\_{CU}[c]
&\ge minOutputShare[a,c] \cdot ActualTotalEffOutputCU[a,r,t] \\\\
OutputSpec[a,c,r,t] \cdot factor\_{CU}[c]
&\le maxOutputShare[a,c] \cdot ActualTotalEffOutputCU[a,r,t]
\end{aligned}
\\]
## C. Full Model Construction
> Note: This section includes references to many features that are not described elsewhere in this
> document or implemented yet (e.g. region-to-region trade), but these are included for
> completeness. This represents the roadmap for future MUSE2 development.
This section describes how all preceding components are integrated to form the complete dispatch
optimisation problem.
### C.1. Objective Function
The overall objective is to minimise the total system cost, which is the sum of all operational
costs from assets (standard and flexible), financial impacts from policy scopes (taxes minus
credits), costs of inter-regional trade, costs of pool-based trade, and importantly, the high
economic penalties associated with any unserved demand for critical commodities:
\\[
\begin{aligned}
\text{Minimise: } &(\text{Core Asset Operational Costs from A.3 and B.4}) \\\\
&+ \sum_{c \in \mathbf{C}^{VoLL},r,t} UnmetD[c,r,t] \cdot VoLL[c,r]
\quad \text{(Penalty for Unserved Demand)}
\end{aligned}
\\]
Note that the unmet demand variables (\\( UnmetD[c,r,t] \\)) are normally not included in the
optimisation and are currently only used to diagnose the source of errors when running the model.
### C.2. Constraints
The complete set of constraints that the optimisation must satisfy includes:
- Capacity & Availability constraints for all assets \\( a \in \mathbf{A} \\)
(as per [A.4] and [B.5])
- [Flexible Asset operational constraints][B.5]
[A.4]: #a4-constraints-capacity--availability-for-standard-assets--a-in-mathbfastd-
[B.5]: #b5-constraints-for--a-in-mathbfaflex-r-in-mathbfr-t-in-mathbft-
#### Demand Satisfaction for \\( c\in \mathbf{C}^{\mathrm{SVD}} \\)
These constraints ensure that exogenously defined final demands for SVDs are met in each region \\(
r \\) and time slice \\( t \\), or any shortfall is explicitly accounted for.
For all \\( r,t,c \in \mathbf{C}^{\mathrm{SVD}} \\): Let \\( TotalSystemProduction_{SVD}[c,r,t] \\)
be the sum of all production of \\( c \\) from standard assets (\\( output_{coeff}[a,c]\\,act[a,r,t]
\\)) and flexible assets (the relevant \\( OutputSpec[a,c,r,t] \\) if \\( c \in
\mathbf{C}\_a^{eff\\_out} \\), or \\( act[a,r,t] \cdot coeff\_{aux\\_out}[a,c] \\) if \\( c \in
\mathbf{C}^{aux\\_out}\_a \\)).
Let \\( NetImports_{SVD}[c,r,t] \\) be net imports of \\( c \\) from R2R and Pool trade if SVDs are
tradeable. If \\( c \in \mathbf{C}^{VoLL} \\) (meaning unserved demand for this SVD is permitted at
a penalty):
\\[
TotalSystemProduction_{SVD}[c,r,t] + NetImports_{SVD}[c,r,t] + UnmetD[c,r,t]
\ge demand[r,c] \times timeslice\\_ share[c,t]
\\]
Else (if SVD \\( c \\) must be strictly met and is not included in \\( \mathbf{C}^{VoLL} \\)):
\\[
TotalSystemProduction_{SVD}[c,r,t] + NetImports_{SVD}[c,r,t]
\ge demand[r,c] \times timeslice\\_ share[c,t]
\\]
#### Commodity Balance for \\( c\in \mathbf{C}^{\mathrm{SED}} \\)
These constraints ensure that for all intermediate SED commodities, total supply equals total demand
within each region \\( r \\) and for each balancing period defined by \\( balance\\_ level[c,r] \\)
(e.g., timeslice, seasonal, annual).
For a timeslice balance (\\( \forall r,t,c \in \mathbf{C}^{\mathrm{SED}} \\)):
Total Inflows (Local Production by all assets + Imports from other regions and pools + Unserved SED
if \\( c \in \mathbf{C}^{VoLL} \\)) = Total Outflows (Local Consumption by all assets + Exports to
other regions).
\\[
\begin{aligned}
&\sum\_{a \in \mathbf{A}^{std}} output_{coeff}[a,c]\\,act[a,r,t]
&& \text{(Std Asset Production)} \\\\
&+ \sum\_{a \in \mathbf{A}^{flex}}
\left(
\begin{cases}
OutputSpec[a,c,r,t] & \text{if } c \in \mathbf{C}^{eff\\_out}\_a \\\\
act[a,r,t] \cdot coeff\_{aux\\_out}[a,c] & \text{if } c \in \mathbf{C}^{aux\\_out}\_a \\\\
0 & \text{otherwise}
\end{cases}
\right)
&& \text{(Flex Asset Production)} \\\\
&-\sum\_{a \in \mathbf{A}^{std}} input\_{coeff}[a,c]\\,act[a,r,t]
&& \text{(Std Asset Consumption)} \\\\
&- \sum\_{a \in \mathbf{A}^{flex}}
\left(
\begin{cases}
InputSpec[a,c,r,t] & \text{if } c \in \mathbf{C}^{eff\\_in}\_a \\\\
act[a,r,t] \cdot coeff\_{aux\\_in}[a,c] & \text{if } c \in \mathbf{C}^{aux\\_in}\_a \\\\
0 & \text{otherwise}
\end{cases}
\right)
&& \text{(Flex Asset Consumption)} \\\\
&+ \sum\_{r'\neq r, c \in \mathbf{C}^R} ship\_{R2R}[r',r,c,t](1 - loss\_{R2R}[r',r,c,t])
&& \text{(R2R Imports)} \\\\
&+ \sum\_{p, c \in \mathbf{C}^P} ship\_{pool}[p,r,c,t](1 - loss\_{pool}[p,r,c,t])
&& \text{(Pool Imports)} \\\\
&- \sum\_{r'\neq r, c \in \mathbf{C}^R} ship\_{R2R}[r,r',c,t]
&& \text{(R2R Exports)} \\\\
&+ \mathbb{I}(c \in \mathbf{C}^{VoLL}) \cdot UnmetD[c,r,t]
&& \text{(Unserved SED, if modelled)} \\\\
&\ge 0
\end{aligned}
\\]
(where \\( \mathbb{I}(c \in \mathbf{C}^{VoLL}) \\) is an indicator function, \\( 1 \\) if \\( c \\)
is in \\( \mathbf{C}^{VoLL} \\), \\( 0 \\) otherwise. Note that SVDs are not consumed by assets, so
\\( input_{coeff}[a,c] \\) and related terms for SVDs on the consumption side are zero).