Constraints

All constraints are grouped by topic. Each equation name matches the label in base.gms. For notation and variable definitions, see Objective Function and Reference.

[Dispatch mode only] — active when fDispatchMode=1. [Optional] — activated by a flag in pSettings.csv.

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Demand and Supply Balance

The demand balance (eDemSupply) equates demand to the composite supply variable:

\[ pDemandData_{z,q,d,y,t} \cdot pEnergyEfficiencyFactor_{z,y} = vSupply_{z,q,d,t,y}. \]

vSupply decomposes generation, storage, flows, trade, and slack (eDefineSupply):

\[ \begin{aligned} vSupply_{z,q,d,t,y} ={}& \sum_{g,f} vPwrOut_{g,f,q,d,t,y} - \sum_{z_2} vFlow_{z,z_2,q,d,t,y} \\ &+ \sum_{z_2} \bigl(1 - pLossFactorInternal_{z,z_2,y}\bigr) \cdot vFlow_{z_2,z,q,d,t,y} \\ &- \sum_{st \in z} vStorInj_{st,q,d,t,y} \cdot \mathbb{1}_{fEnableStorage} \\ &+ \sum_{z_{\text{ext}}} vYearlyImportExternal_{z,z_{\text{ext}},q,d,t,y} - \sum_{z_{\text{ext}}} vYearlyExportExternal_{z,z_{\text{ext}},q,d,t,y} \\ &+ vUSE_{z,q,d,t,y} - vSurplus_{z,q,d,t,y}. \end{aligned} \]

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Capacity Evolution

Initial capacity for the first model year, where the installed capacity term applies only to existing generators (eg) commissioned before the horizon (eInitialCapacity):

\[ vCap_{g,y_0} = pGenData_{g,\text{Capacity}} \cdot \mathbb{1}_{eg(g)} + vBuild_{g,y_0} - vRetire_{g,y_0}. \]

For subsequent years, existing assets follow (eCapacityEvolutionExist):

\[ vCap_{eg,y} = vCap_{eg,y-1} + vBuild_{eg,y} - vRetire_{eg,y}. \]

New entrants have no retirement (eCapacityEvolutionNew):

\[ vCap_{ng,y} = vCap_{ng,y-1} + vBuild_{ng,y}. \]

Discrete build and retirement decisions are enforced through integer variables (eBuiltCap, eRetireCap):

\[ vBuild_{ng,y} = pGenData_{ng,\text{UnitSize}} \cdot vBuiltCapVar_{ng,y}, \qquad vRetire_{eg,y} = pGenData_{eg,\text{UnitSize}} \cdot vRetireCapVar_{eg,y}. \]

Total builds for existing-fleet generators commissioned after the model start are capped (eInitialBuildLimit):

\[ \sum_y vBuild_{eg,y} \le pGenData_{eg,\text{Capacity}}. \]

Storage and CSP thermal capacity follow analogous recursions (eCapStor*, eCapTherm*).

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Generator Operating Limits

Dispatch and Reserve Envelope

Combined output and spinning reserve cannot exceed installed capacity (eJointResCap):

\[ \sum_f vPwrOut_{g,f,q,d,t,y} + vSpinningReserve_{g,q,d,t,y} \le \bigl(1 + pGenData_{g,\text{Overloadfactor}}\bigr) \cdot vCap_{g,y}. \]

Seasonal Availability

Quarterly energy output is limited by the seasonal availability factor (eMaxCF). Note that pAvailability carries a year index to allow for degradation or scheduled outages:

\[ \sum_{f,d,t} vPwrOut_{g,f,q,d,t,y} \cdot pHours_{q,d,t} \le pAvailability_{g,y,q} \cdot vCap_{g,y} \cdot \sum_{d,t} pHours_{q,d,t}. \]

Minimum Output [Optional — fApplyMinGenShareAllHours]

(eMinGen):

\[ \sum_f vPwrOut_{g,f,q,d,t,y} \ge vCap_{g,y} \cdot pGenData_{g,\text{MinGenShareAllHours}}. \]

Renewable Generation with Curtailment

VRE output follows the hourly profile scaled by seasonal availability (eVREProfile). Curtailment is tracked explicitly:

\[ vPwrOut_{g,f,q,d,t,y} + vCurtailedVRE_{z,g,q,d,t,y} = pVREgenProfile_{g,q,d,t} \cdot pAvailability_{g,y,q} \cdot vCap_{g,y}. \]

Curtailment is penalised through vYearlyCurtailmentCost in the objective function.

Fuel Consumption

(eFuel):

\[ vFuel_{z,f,y} = \sum_{g \in z,\, q,d,t} vPwrOut_{g,f,q,d,t,y} \cdot pHours_{q,d,t} \cdot pHeatRate_{g,f}. \]

An upper bound on annual fuel consumption by country and fuel applies via eFuelLimit when fApplyFuelConstraint=1.

Generation Phase-out [Optional — fApplyGenerationPhaseout]

eMaxGenerationByFuel caps annual generation by zone, technology group, and fuel (in GWh):

\[ \frac{1}{1000} \sum_{g \in tech \cap z,\, q,d,t} vPwrOut_{g,f,q,d,t,y} \cdot pHours_{q,d,t} \le pMaxGenerationByFuel_{z,tech,f,y}. \]

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Ramp Rate Constraints

Standard mode

Inter-hour ramp limits apply within each representative day (eRampUpLimit, eRampDnLimit):

\[ \sum_f vPwrOut_{g,f,q,d,t,y} - \sum_f vPwrOut_{g,f,q,d,t-1,y} \le vCap_{g,y} \cdot pGenData_{g,\text{RampUpRate}}, \]

\[ \sum_f vPwrOut_{g,f,q,d,t-1,y} - \sum_f vPwrOut_{g,f,q,d,t,y} \le vCap_{g,y} \cdot pGenData_{g,\text{RampDnRate}}. \]

Analogous bounds apply to storage charging (eChargeRampUpLimit, eChargeRampDownLimit).

Dispatch mode [Dispatch mode only]

In dispatch mode, ramp limits operate over chronological slots \(AT\) using vPwrOutDispatch, linked to vPwrOut via eRampContinuity. Startup and shutdown events provide additional headroom when unit commitment is active (eDispatchRampUpLimit, eDispatchRampDownLimit):

\[ vPwrOutDispatch_{g,AT,y} - vPwrOutDispatch_{g,AT-1,y} \le vCap_{g,y} \cdot pGenData_{g,\text{RampUpRate}} + isStartup_{g,AT,y} \cdot pGenData_{g,\text{Capacity}} \cdot \mathbb{1}_{fApplyMinGenCommitment}, \]

\[ vPwrOutDispatch_{g,AT-1,y} - vPwrOutDispatch_{g,AT,y} \le vCap_{g,y} \cdot pGenData_{g,\text{RampDnRate}} + isShutdown_{g,AT,y} \cdot pGenData_{g,\text{Capacity}} \cdot \mathbb{1}_{fApplyMinGenCommitment}. \]

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Reserve Requirements

Generator reserve limit

Maximum spinning reserve provision per generator (eSpinningReserveLim):

\[ vSpinningReserve_{g,q,d,t,y} \le vCap_{g,y} \cdot pGenData_{g,\text{ResLimShare}}. \]

For VRE generators the limit is further scaled by the hourly profile (eSpinningReserveLimVRE):

\[ vSpinningReserve_{g,q,d,t,y} \le vCap_{g,y} \cdot pGenData_{g,\text{ResLimShare}} \cdot pVREgenProfile_{g,q,d,t}. \]

Spinning reserve requirement [Optional]

Country level (eSpinningReserveReqCountry, flag fApplyCountrySpinReserveConstraint):

\[ \sum_{z \in c,\, g \in z} vSpinningReserve_{g,q,d,t,y} + vUnmetSpinningReserveCountry_{c,q,d,t,y} \ge pSpinningReserveReqCountry_{c,y} + \sum_{g \in \text{VREnoROR},\, f} vPwrOut_{g,f,q,d,t,y} \cdot psVREForecastErrorPct. \]

System level (eSpinningReserveReqSystem, flag fApplySystemSpinReserveConstraint):

\[ \sum_{g} vSpinningReserve_{g,q,d,t,y} + vUnmetSpinningReserveSystem_{q,d,t,y} \ge pSpinningReserveReqSystem_{y} + \sum_{g \in \text{VREnoROR},\, f} vPwrOut_{g,f,q,d,t,y} \cdot psVREForecastErrorPct. \]

Planning reserve margin [Optional — fApplyPlanningReserveConstraint]

Country level (ePlanningReserveReqCountry). When fCountIntercoForReserves=1, available interconnection capacity contributes to the margin:

\[ \sum_{z \in c,\, g \in z} vCap_{g,y} \cdot pCapacityCredit_{g,y} + vUnmetPlanningReserveCountry_{c,y} + IC_{c,y} \ge \bigl(1 + pPlanningReserveMargin_{c}\bigr) \cdot \max_{q,d,t} \sum_{z \in c} pDemandData_{z,q,d,y,t} \cdot pEnergyEfficiencyFactor_{z,y}, \]

where the interconnection term \(IC_{c,y}\) (zero when fCountIntercoForReserves=0) is:

\[ IC_{c,y} = \sum_{z \in c,\, z_2 \notin c} \left( \frac{\sum_q pTransferLimit_{z_2,z,q,y}}{|Q|} + vNewTransmissionLine_{z_2,z,y} \cdot pNewTransmission_{z_2,z,\text{CapacityPerLine}} \cdot fAllowTransferExpansion \right). \]

System level (ePlanningReserveReqSystem):

\[ \sum_{g} vCap_{g,y} \cdot pCapacityCredit_{g,y} + vUnmetPlanningReserveSystem_{y} \ge \bigl(1 + sReserveMarginPct\bigr) \cdot \max_{q,d,t} \sum_z pDemandData_{z,q,d,y,t} \cdot pEnergyEfficiencyFactor_{z,y}. \]

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Transmission and Trade

Internal transfer capacity (eTransferCapacityLimit):

\[ vFlow_{z,z_2,q,d,t,y} \le pTransferLimit_{z,z_2,q,y} + vNewTransmissionLine_{z,z_2,y} \cdot pNewTransmission_{z,z_2,\text{CapacityPerLine}} \cdot fAllowTransferExpansion. \]

Expansion symmetry and cumulative build tracking: eCumulativeTransferExpansion, eSymmetricTransferBuild.

External exchange limits [Optional]: annual and hourly import/export share caps (eMaxAnnualImportShareEnergy, eMaxAnnualExportShareEnergy, eMaxHourlyImportShareEnergy, eMaxHourlyExportShareEnergy), point-to-point limits (eExternalImportLimit, eExternalExportLimit), and a minimum import floor (eMinImportRequirement) when pMinImport > 0.

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Storage Operations

Capacity and power bounds

State-of-charge upper bound (eSOCUpperBound):

\[ vStorage_{st,q,d,t,y} \le vCapStor_{st,y}. \]

Power injection and energy-power consistency (eChargeCapacityLimit, eStorageCapMinConstraint):

\[ vStorInj_{st,q,d,t,y} \le vCap_{st,y}, \qquad vCapStor_{st,y} \ge vCap_{st,y}. \]

When a fixed energy-to-power ratio is specified (eStorageFixedDuration):

\[ vCapStor_{st,y} = vCap_{st,y} \cdot pStorageData_{st,\text{StorageDuration}}. \]

Net charge balance (eNetChargeBalance):

\[ vStorNet_{st,q,d,t,y} = \sum_f vPwrOut_{st,f,q,d,t,y} - vStorInj_{st,q,d,t,y}. \]

State-of-charge dynamics

Three equations govern SOC evolution depending on time position and run mode.

First hour of a representative day (eStateOfChargeInitRep) — standard mode only, no carry-over from the previous day:

\[ vStorage_{st,q,d,t_1,y} = pStorageData_{st,\text{Efficiency}} \cdot vStorInj_{st,q,d,t_1,y} - \sum_f vPwrOut_{st,f,q,d,t_1,y}. \]

All subsequent hours (eStorageHourTransition):

\[ vStorage_{st,q,d,t,y} = vStorage_{st,q,d,t-1,y} + pStorageData_{st,\text{Efficiency}} \cdot vStorInj_{st,q,d,t,y} - \sum_f vPwrOut_{st,f,q,d,t,y}. \]

Day boundary — [Dispatch mode only] (eStorageDayWrap) — links the first hour of each day to the last chronological slot \(AT-1\):

\[ vStorage_{st,q,d,t_1,y} = vStorage_{st,AT-1,y} + pStorageData_{st,\text{Efficiency}} \cdot vStorInj_{st,q,d,t_1,y} - \sum_f vPwrOut_{st,f,q,d,t_1,y}. \]

Cycle anchoring — [Dispatch mode only] (eStorageSOCInitDispatch, eStorageSOCFinalDispatch) — the first and last hours of the full chronological cycle are pinned to a prescribed share of capacity:

\[ vStorage_{st,q,d,t,y} = vCapStor_{st,y} \cdot pStorageInitShare. \]

Reserve contribution

Storage can provide spinning reserve up to its current state of charge (eSOCSupportsReserve):

\[ vSpinningReserve_{st,q,d,t,y} \le vStorage_{st,q,d,t,y}. \]

PV-coupled batteries are additionally capped by the paired PV profile (eChargeLimitWithPVProfile).

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CSP Extensions

CSP plants use dedicated variables for the solar thermal field (vThermalOut, vCapTherm) alongside the standard storage variables.

Charging is limited to what the thermal field produces, scaled by its efficiency (eCSPStorageInjectionLimit):

\[ vStorInj_{cs,q,d,t,y} \le vThermalOut_{cs,q,d,t,y} \cdot \eta_{\text{thermal field}}. \]

Thermal output is bounded by field capacity and the solar resource profile (eCSPThermalOutputLimit):

\[ vThermalOut_{cs,q,d,t,y} \le vCapTherm_{cs,y} \cdot pCSPProfile_{cs,q,d,t}. \]

The power balance links thermal output, storage charge/discharge, and turbine dispatch (eCSPPowerBalance):

\[ vThermalOut_{cs,q,d,t,y} \cdot \eta_{\text{thermal}} - vStorInj_{cs,q,d,t,y} + vStorOut_{cs,q,d,t,y} \cdot \eta_{\text{storage}} = \sum_f vPwrOut_{cs,f,q,d,t,y}. \]

Storage capacity and charging are also bounded by eCSPStorageCapacityLimit and eCSPStorageInjectionCap. SOC dynamics follow the same recursion as standard storage (eCSPStorageEnergyBalance, eCSPStorageInitialBalance). Capacity evolution for both subsystems uses eCapStor* and eCapTherm*.

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Investment and Policy

Capital budget [Optional — fApplyCapitalConstraint] (eCapitalConstraint):

\[ \sum_{y,\, ng} pRR_y \cdot pWeightYear_y \cdot pCRF_{ng} \cdot vCap_{ng,y} \cdot pGenData_{ng,\text{Capex}} \le sMaxCapitalInvestmentInvestment \times 10^3. \]

Minimum renewable share [Optional] (eMinGenRE): a fraction pMinRE of total annual energy must come from renewable sources.

Fuel availability cap [Optional — fApplyFuelConstraint] (eFuelLimit): annual consumption by country and fuel cannot exceed pMaxFuelLimit.

Minimum import [Optional] (eMinImportRequirement): enforces a lower bound on imports from a given zone when pMinImport > 0.

Discrete investment decisions for transmission (vBuildTransmissionLine) and generators/retirements (vBuiltCapVar, vRetireCapVar) are integer variables bounded by capacity consistency equations.

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CO₂ Emissions

Zonal emissions are computed from dispatched generation, heat rates, and carbon content (eZonalEmissions):

\[ vZonalEmissions_{z,y} = \sum_{g \in z,\, f,\, q,d,t} vPwrOut_{g,f,q,d,t,y} \cdot pHours_{q,d,t} \cdot pHeatRate_{g,f} \cdot pFuelCarbonContent_f. \]

System total (eTotalEmissions):

\[ vTotalEmissions_y = \sum_z vZonalEmissions_{z,y}. \]

Country cap [Optional — fApplyCountryCo2Constraint] (eEmissionsCountry):

\[ \sum_{z \in c} vZonalEmissions_{z,y} - vYearlyCO2backstop_{c,y} \le pEmissionsCountry_{c,y}. \]

System cap [Optional — fApplySystemCo2Constraint] (eTotalEmissionsConstraint):

\[ vTotalEmissions_y - vYearlySysCO2backstop_y \le pEmissionsTotal_y. \]

The backstop slack variables allow the cap to be violated at a penalty cost, which feeds into the objective function via vYearlyCO2BackstopCostCountry and vYearlyCO2BackstopCostSystem.

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Unit Commitment [Dispatch mode only]

Unit commitment is active when fDispatchMode=1 and fApplyMinGenCommitment=1. Binary variables isOn, isStartup, and isShutdown track generator state at each chronological slot \(AT\).

Transition consistency (eCommitmentConsistency):

\[ isOn_{g,AT,y} - isOn_{g,AT-1,y} = isStartup_{g,AT,y} - isShutdown_{g,AT,y}. \]

eCommitmentInitialization sets the state at the first slot; eCommitmentSingleTransition prevents simultaneous startup and shutdown.

Min/max generation tied to commitment state (eDispatchMinGenPoint, eDispatchMaxGenPoint):

\[ vCap_{g,y} \cdot pGenData_{g,\text{MinGenShareAllHours}} \cdot isOn_{g,AT,y} \le vPwrOutDispatch_{g,AT,y} \le vCap_{g,y} \cdot (1 + pGenData_{g,\text{Overloadfactor}}) \cdot isOn_{g,AT,y}. \]

Minimum up/down time [Optional — fApplyMUDT]: eMinUpInitial, eMinUpRolling, eMinDownInitial, eMinDownRolling prevent generators from cycling faster than their physical limits.

Startup costs [Optional — fApplyStartupCost]: eStartupCostConstraint accumulates per-slot startup costs into vYearlyStartupCost, which enters the objective via vYearlyVariableCost.