2b_equations.gms 91.7 KB
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This file is part of Backbone.

Backbone is free software: you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

Backbone is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public License
along with Backbone.  If not, see <http://www.gnu.org/licenses/>.
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* =============================================================================
* --- Penalty Definitions -----------------------------------------------------
* =============================================================================

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$setlocal def_penalty 1e4
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Scalars
    PENALTY "Default equation violation penalty" / %def_penalty% /
;
Parameters
    PENALTY_BALANCE(grid) "Penalty on violating energy balance eq. (EUR/MWh)"
    PENALTY_RES(restype, up_down) "Penalty on violating a reserve (EUR/MW)"
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    PENALTY_RES_MISSING(restype, up_down) "Penalty on violating a reserve (EUR/MW)"
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;
PENALTY_BALANCE(grid) = %def_penalty%;
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PENALTY_RES(restype, up_down) = 0.9*%def_penalty%;
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PENALTY_RES_MISSING(restype, up_down) = 0.1*%def_penalty%;
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* =============================================================================
* --- Equation Declarations ---------------------------------------------------
* =============================================================================

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equations
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    // Objective Function, Energy Balance, and Reserve demand
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    q_obj "Objective function"
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    q_balance(grid, node, mType, f, t) "Energy demand must be satisfied at each node"
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    q_resDemand(restype, up_down, node, f, t) "Procurement for each reserve type is greater than demand"
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    // Unit Operation
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    q_maxDownward(mType, grid, node, unit, f, t) "Downward commitments will not undercut power plant minimum load constraints or maximum elec. consumption"
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    q_noReserveInRunUp(mType, grid, node, unit, f, t)
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    q_maxUpward(mType, grid, node, unit, f, t) "Upward commitments will not exceed maximum available capacity or consumed power"
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    q_startshut(mType, unit, f, t) "Online cap. now minus online cap in the previous time step is equal to started up minus shut down capacity"
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    q_startuptype(mType, starttype, unit, f, t) "Startup type depends on the time the unit has been non-operational"
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    q_onlineOnStartUp(unit, f, t) "Unit must be online after starting up"
    q_offlineAfterShutdown(unit, f, t) "Unit must be offline after shutting down"
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    q_onlineLimit(mType, unit, f, t) "Number of online units limited for units with startup constraints and investment possibility"
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    q_onlineMinUptime(mType, unit, f, t) "Unit must stay operational if it has started up during the previous minOperationHours hours"
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    q_genRamp(mType, grid, node, s, unit, f, t) "Record the ramps of units with ramp restricitions or costs"
    q_rampUpLimit(mType, grid, node, s, unit, f, t) "Up ramping limited for units"
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    q_rampDownLimit(grid, node, mType, s, unit, f, t) "Down ramping limited for units"
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    q_outputRatioFixed(grid, node, grid, node, unit, f, t) "Force fixed ratio between two energy outputs into different energy grids"
    q_outputRatioConstrained(grid, node, grid, node, unit, f, t) "Constrained ratio between two grids of energy output; e.g. electricity generation is greater than cV times unit_heat generation in extraction plants"
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    q_conversionDirectInputOutput(effSelector, unit, f, t) "Direct conversion of inputs to outputs (no piece-wise linear part-load efficiencies)"
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    q_conversionSOS2InputIntermediate(effSelector, unit, f, t)   "Intermediate output when using SOS2 variable based part-load piece-wise linearization"
    q_conversionSOS2Constraint(effSelector, unit, f, t)          "Sum of v_sos2 has to equal v_online"
    q_conversionSOS2IntermediateOutput(effSelector, unit, f, t)  "Output is forced equal with v_sos2 output"
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    // Energy Transfer
    q_transfer(grid, node, node, f, t) "Rightward and leftward transfer must match the total transfer"
    q_transferRightwardLimit(grid, node, node, f, t) "Transfer of energy and capacity reservations to the rightward direction are less than the transfer capacity"
    q_transferLeftwardLimit(grid, node, node, f, t) "Transfer of energy and capacity reservations to the leftward direction are less than the transfer capacity"
    q_resTransferLimitRightward(grid, node, node, f, t) "Transfer of energy and capacity reservations are less than the transfer capacity to the rightward direction"
    q_resTransferLimitLeftward(grid, node, node, f, t) "Transfer of energy and capacity reservations are less than the transfer capacity to the leftward direction"

    // State Variables
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    q_stateSlack(grid, node, slack, f, t) "Slack variable greater than the difference between v_state and the slack boundary"
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    q_stateUpwardLimit(grid, node, mType, f, t) "Limit the commitments of a node with a state variable to the available headrooms"
    q_stateDownwardLimit(grid, node, mType, f, t) "Limit the commitments of a node with a state variable to the available headrooms"
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*    q_boundState(grid, node, node, mType, f, t) "Node state variables bounded by other nodes"
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    q_boundStateMaxDiff(grid, node, node, mType, f, t) "Node state variables bounded by other nodes (maximum state difference)"
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    q_boundCyclic(grid, node, mType, s, s) "Cyclic bound for the first and the last states of samples"
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*    q_boundCyclicSamples(grid, node, mType, s, f, t, s_, f_, t_) "Cyclic bound inside or between samples"
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    // Policy
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    q_inertiaMin(group, f, t) "Minimum inertia in a group of nodes"
    q_instantaneousShareMax(group, f, t) "Maximum instantaneous share of generation and controlled import from a group of units and links"
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    q_capacityMargin(grid, node, f, t) "There needs to be enough capacity to cover energy demand plus a margin"
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    q_constrainedCapMultiUnit(group, t) "Constrained unit number ratios and sums for a group of units"
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    q_emissioncap(group, emission) "Limit for emissions"
    q_energyShareMax(group) "Maximum energy share of generation and import from a group of units"
    q_energyShareMin(group) "Minimum energy share of generation and import from a group of units"
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;

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* =============================================================================
* --- Equation Definitions ----------------------------------------------------
* =============================================================================

* --- Objective Function ------------------------------------------------------
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q_obj ..
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    + v_obj * 1e6
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    =E=

    // Sum over all the samples, forecasts, and time steps in the current model
    + sum(msft(m, s, f, t),
        // Probability (weight coefficient) of (s,f,t)
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        + p_msft_probability(m, s, f, t)
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            * [
                // Time step length dependent costs
                + p_stepLength(m, f, t)
                    * [
                        // Variable O&M costs
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                        + sum(gnuft(gnu_output(grid, node, unit), f, t),  // Calculated only for output energy
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                            + v_gen(grid, node, unit, f, t)
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                                * p_unit(unit, 'omCosts')
                            ) // END sum(gnu_output)

                        // Fuel and emission costs
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                        + sum(uFuel(unit, 'main', fuel)${uft(unit, f, t)},
                            + v_fuelUse(fuel, unit, f, t)
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                                * [
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                                    + ts_fuelPrice_(fuel ,t)
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                                    + sum(emission, // Emission taxes
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                                        + p_unitFuelEmissionCost(unit, fuel, emission)
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                                        )
                                    ] // END * v_fuelUse
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                            ) // END sum(uFuel)
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                        // Node state slack variable costs
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                        + sum(gn_stateSlack(grid, node),
                            + sum(slack${p_gnBoundaryPropertiesForStates(grid, node, slack, 'slackCost')},
                                + v_stateSlack(grid, node, slack, f, t)
                                    * p_gnBoundaryPropertiesForStates(grid, node, slack, 'slackCost')
                                ) // END sum(slack)
                            ) // END sum(gn_stateSlack)

                        // Dummy variable penalties
                        // Energy balance feasibility dummy varible penalties
                        + sum(inc_dec,
                            + sum(gn(grid, node),
                                + vq_gen(inc_dec, grid, node, f, t)
                                    * PENALTY_BALANCE(grid)
                                ) // END sum(gn)
                            ) // END sum(inc_dec)

                        // Reserve provision feasibility dummy variable penalties
                        + sum(restypeDirectionNode(restype, up_down, node),
                            + vq_resDemand(restype, up_down, node, f, t)
                                * PENALTY_RES(restype, up_down)
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                            + vq_resMissing(restype, up_down, node, f, t)$(ord(t) <= tSolveFirst + p_nReserves(node, restype, 'gate_closure') - mod(tSolveFirst - 1, p_nReserves(node, restype, 'update_frequency')))
                                * PENALTY_RES_MISSING(restype, up_down)
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                            ) // END sum(restypeDirectionNode)

                        ] // END * p_stepLength

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                // Start-up costs, initial startup free as units could have been online before model started
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                + sum(uft_online(unit, f, t),
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                    + sum(unitStarttype(unit, starttype),
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                        + v_startup(unit, starttype, f+df_central(f,t), t) // Cost of starting up
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                            * [ // Startup variable costs
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                                + p_uStartup(unit, starttype, 'cost', 'unit')
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                                // Start-up fuel and emission costs
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                                + sum(uFuel(unit, 'startup', fuel),
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                                    + p_uStartup(unit, starttype, 'consumption', 'unit')  //${ not unit_investLP(unit) }  WHY THIS CONDITIONAL WOULD BE NEEDED?
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                                        * [
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                                            + ts_fuelPrice_(fuel, t)
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                                            + sum(emission, // Emission taxes of startup fuel use
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                                                + p_unitFuelEmissionCost(unit, fuel, emission)
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                                              ) // END sum(emission)
                                          ] // END * p_uStartup
                                  ) // END sum(uFuel)
                              ] // END * v_startup
                      ) // END sum(starttype)
                  ) // END sum(uft_online)
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                // !!! PENDING CHANGES !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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                // Ramping costs
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                + sum(gnuft_ramp(grid, node, unit, f, t)${  p_gnu(grid, node, unit, 'rampUpCost')
                                                            or p_gnu(grid, node, unit, 'rampDownCost')
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                                                            },
                    + p_gnu(grid, node, unit, 'rampUpCost') * v_genRampChange(grid, node, unit, 'up', f, t)
                    + p_gnu(grid, node, unit, 'rampDownCost') * v_genRampChange(grid, node, unit, 'down', f, t)
                    ) // END sum(gnuft_ramp)
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                ]  // END * p_sft_probability(s,f,t)
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        ) // END sum over msft(m, s, f, t)

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    // Cost of energy storage change
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    + sum(gn_state(grid, node),
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        + sum(mft_start(m, f, t)${  p_storageValue(grid, node, t)
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                                    and active(m, 'storageValue')
                                    },
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            + v_state(grid, node, f, t)
                * p_storageValue(grid, node, t)
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                * sum(ms(m, s)${ p_msft_probability(m, s, f, t) },
                    + p_msft_probability(m, s, f, t)
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                    ) // END sum(s)
            ) // END sum(mftStart)
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        - sum(mft_lastSteps(m, f, t)${  p_storageValue(grid, node, t)
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                                        and active(m, 'storageValue')
                                        },
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            + v_state(grid, node, f, t)
                * p_storageValue(grid, node, t)
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                * sum(ms(m, s)${p_msft_probability(m, s, f, t)},
                    + p_msft_probability(m, s, f, t)
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                    ) // END sum(s)
            ) // END sum(mftLastSteps)
        ) // END sum(gn_state)

    // Investment Costs
    + sum(t_invest(t),
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        // Unit investment costs (including fixed operation and maintenance costs)
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        + sum(gnu(grid, node, unit),
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            + v_invest_LP(unit, t)${ unit_investLP(unit) }
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                * p_gnu(grid, node, unit, 'unitSizeTot')
                * [
                    + p_gnu(grid, node, unit, 'invCosts') * p_gnu(grid, node, unit, 'annuity')
                    + p_gnu(grid, node, unit, 'fomCosts')
                  ]
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            + v_invest_MIP(unit, t)${ unit_investMIP(unit) }
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                * p_gnu(grid, node, unit, 'unitSizeTot')
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                * [
                    + p_gnu(grid, node, unit, 'invCosts') * p_gnu(grid, node, unit, 'annuity')
                    + p_gnu(grid, node, unit, 'fomCosts')
                  ]
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            ) // END sum(gnu)

        // Transfer link investment costs
        + sum(gn2n_directional(grid, from_node, to_node),
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            + v_investTransfer_LP(grid, from_node, to_node, t)${ not p_gnn(grid, from_node, to_node, 'investMIP') }
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                * [
                    + p_gnn(grid, from_node, to_node, 'invCost')
                        * p_gnn(grid, from_node, to_node, 'annuity')
                    + p_gnn(grid, to_node, from_node, 'invCost')
                        * p_gnn(grid, to_node, from_node, 'annuity')
                    ] // END * v_investTransfer_LP
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            + v_investTransfer_MIP(grid, from_node, to_node, t)${ p_gnn(grid, from_node, to_node, 'investMIP') }
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                * [
                    + p_gnn(grid, from_node, to_node, 'unitSize')
                        * p_gnn(grid, from_node, to_node, 'invCost')
                        * p_gnn(grid, from_node, to_node, 'annuity')
                    + p_gnn(grid, to_node, from_node, 'unitSize')
                        * p_gnn(grid, to_node, from_node, 'invCost')
                        * p_gnn(grid, to_node, from_node, 'annuity')
                    ] // END * v_investTransfer_MIP
            ) // END sum(gn2n_directional)
        ) // END sum(t_invest)
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;
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* --- Energy Balance ----------------------------------------------------------

q_balance(gn(grid, node), mft(m, f, t))${   not p_gn(grid, node, 'boundAll')
                                            } .. // Energy/power balance dynamics solved using implicit Euler discretization

    // The left side of the equation is the change in the state (will be zero if the node doesn't have a state)
    + p_gn(grid, node, 'energyStoredPerUnitOfState')${gn_state(grid, node)} // Unit conversion between v_state of a particular node and energy variables (defaults to 1, but can have node based values if e.g. v_state is in Kelvins and each node has a different heat storage capacity)
        * [
            + v_state(grid, node, f+df_central(f,t), t)                   // The difference between current
            - v_state(grid, node, f+df(f,t+dt(t)), t+dt(t))                     // ... and previous state of the node
            ]

    =E=

    // The right side of the equation contains all the changes converted to energy terms
    + p_stepLength(m, f, t) // Multiply with the length of the timestep to convert power into energy
        $$ifi '%rampSched%' == 'yes' / 2    // Averaging all the terms on the right side of the equation over the timestep here.
        * (
            // Self discharge out of the model boundaries
            - p_gn(grid, node, 'selfDischargeLoss')${gn_state(grid, node)}
                * [
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                    + v_state(grid, node, f+df_central(f,t), t) // The current state of the node
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                    $$ifi '%rampSched%' == 'yes' + v_state(grid, node, f+df(f,t+dt(t)), t+dt(t)) // and possibly averaging with the previous state of the node
                    ]

            // Energy diffusion from this node to neighbouring nodes
            - sum(to_node${gnn_state(grid, node, to_node)},
                + p_gnn(grid, node, to_node, 'diffCoeff')
                    * [
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                        + v_state(grid, node, f+df_central(f,t), t)
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                        $$ifi '%rampSched%' == 'yes' + v_state(grid, node, f+df(f,t+dt(t)), t+dt(t))
                        ]
                ) // END sum(to_node)

            // Energy diffusion from neighbouring nodes to this node
            + sum(from_node${gnn_state(grid, from_node, node)},
                + p_gnn(grid, from_node, node, 'diffCoeff')
                    * [
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                        + v_state(grid, from_node, f+df_central(f,t), t) // Incoming diffusion based on the state of the neighbouring node
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                        $$ifi '%rampSched%' == 'yes' + v_state(grid, from_node, f+df(f,t+dt(t)), t+dt(t)) // Ramp schedule averaging, NOTE! State and other terms use different indeces for non-ramp-schedule!
                        ]
                ) // END sum(from_node)

            // Controlled energy transfer, applies when the current node is on the left side of the connection
            - sum(node_${gn2n_directional(grid, node, node_)},
                + (1 - p_gnn(grid, node, node_, 'transferLoss')) // Reduce transfer losses
                    * [
                        + v_transfer(grid, node, node_, f, t)
                        $$ifi '%rampSched%' == 'yes' + v_transfer(grid, node, node_, f, t+dt(t)) // Ramp schedule averaging, NOTE! State and other terms use different indeces for non-ramp-schedule!
                        ]
                + p_gnn(grid, node, node_, 'transferLoss') // Add transfer losses back if transfer is from this node to another node
                    * [
                        + v_transferRightward(grid, node, node_, f, t)
                        $$ifi '%rampSched%' == 'yes' + v_transferRightward(grid, node, node_, f, t+dt(t)) // Ramp schedule averaging, NOTE! State and other terms use different indeces for non-ramp-schedule!
                        ]
                ) // END sum(node_)

            // Controlled energy transfer, applies when the current node is on the right side of the connection
            + sum(node_${gn2n_directional(grid, node_, node)},
                + [
                    + v_transfer(grid, node_, node, f, t)
                    $$ifi '%rampSched%' == 'yes' + v_transfer(grid, node_, node, f, t+dt(t)) // Ramp schedule averaging, NOTE! State and other terms use different indeces for non-ramp-schedule!
                    ]
                - p_gnn(grid, node_, node, 'transferLoss') // Reduce transfer losses if transfer is from another node to this node
                    * [
                        + v_transferRightward(grid, node_, node, f, t)
                        $$ifi '%rampSched%' == 'yes' + v_transferRightward(grid, node_, node, f, t+dt(t)) // Ramp schedule averaging, NOTE! State and other terms use different indeces for non-ramp-schedule!
                        ]
                ) // END sum(node_)

            // Interactions between the node and its units
            + sum(gnuft(grid, node, unit, f, t),
                + v_gen(grid, node, unit, f, t) // Unit energy generation and consumption
                $$ifi '%rampSched%' == 'yes' + v_gen(grid, node, unit, f, t+dt(t))
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                )
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            // Spilling energy out of the endogenous grids in the model
            - v_spill(grid, node, f, t)${node_spill(node)}
            $$ifi '%rampSched%' == 'yes' - v_spill(grid, node, f, t)${node_spill(node)}

            // Power inflow and outflow timeseries to/from the node
            + ts_influx_(grid, node, f, t)   // Incoming (positive) and outgoing (negative) absolute value time series
            $$ifi '%rampSched%' == 'yes' + ts_influx_(grid, node, f, t+dt(t))

            // Dummy generation variables, for feasibility purposes
            + vq_gen('increase', grid, node, f, t) // Note! When stateSlack is permitted, have to take caution with the penalties so that it will be used first
            $$ifi '%rampSched%' == 'yes' + vq_gen('increase', grid, node, f, t+dt(t))
            - vq_gen('decrease', grid, node, f, t) // Note! When stateSlack is permitted, have to take caution with the penalties so that it will be used first
            $$ifi '%rampSched%' == 'yes' - vq_gen('decrease', grid, node, f, t+dt(t))
    ) // END * p_stepLength
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;
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* --- Reserve Demand ----------------------------------------------------------

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q_resDemand(restypeDirectionNode(restype, up_down, node), ft(f, t)) ${   ord(t) < tSolveFirst + sum[mf(m, f), mSettings(m, 't_reserveLength')]
                                                                        and not [ restypeReleasedForRealization(restype)
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                                                                                    and ft_realized(f, t)
                                                                                    ]
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                                                                        } ..
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    // Reserve provision by capable units on this node
    + sum(nuft(node, unit, f, t)${nuRescapable(restype, up_down, node, unit)},
        + v_reserve(restype, up_down, node, unit, f+df_nReserves(node, restype, f, t), t)
        ) // END sum(nuft)

    // Reserve provision to this node via transfer links
    + sum(gn2n_directional(grid, node_, node)${restypeDirectionNode(restype, up_down, node_)},
        + (1 - p_gnn(grid, node_, node, 'transferLoss') )
            * v_resTransferRightward(restype, up_down, node_, node, f+df_nReserves(node_, restype, f, t), t)             // Reserves from another node - reduces the need for reserves in the node
        ) // END sum(gn2n_directional)
    + sum(gn2n_directional(grid, node, node_)${restypeDirectionNode(restype, up_down, node_)},
        + (1 - p_gnn(grid, node, node_, 'transferLoss') )
            * v_resTransferLeftward(restype, up_down, node, node_, f+df_nReserves(node_, restype, f, t), t)             // Reserves from another node - reduces the need for reserves in the node
        ) // END sum(gn2n_directional)

    =G=

    // Demand for reserves
    + ts_reserveDemand_(restype, up_down, node, f, t)${p_nReserves(node, restype, 'use_time_series')}
    + p_nReserves(node, restype, up_down)${not p_nReserves(node, restype, 'use_time_series')}

    // Reserve provisions to another nodes via transfer links
    + sum(gn2n_directional(grid, node, node_)${restypeDirectionNode(restype, up_down, node_)},   // If trasferring reserves to another node, increase your own reserves by same amount
        + v_resTransferRightward(restype, up_down, node, node_, f+df_nReserves(node, restype, f, t), t)
        ) // END sum(gn2n_directional)
    + sum(gn2n_directional(grid, node_, node)${restypeDirectionNode(restype, up_down, node_)},   // If trasferring reserves to another node, increase your own reserves by same amount
        + v_resTransferLeftward(restype, up_down, node_, node, f+df_nReserves(node, restype, f, t), t)
        ) // END sum(gn2n_directional)

    // Reserve demand feasibility dummy variables
    - vq_resDemand(restype, up_down, node, f, t)
383
    - vq_resMissing(restype, up_down, node, f, t)$(ord(t) <= tSolveFirst + p_nReserves(node, restype, 'gate_closure') - mod(tSolveFirst - 1, p_nReserves(node, restype, 'update_frequency')))
384
;
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405
406
407

* --- Maximum Downward Capacity -----------------------------------------------

q_maxDownward(m, gnuft(grid, node, unit, f, t))${   [   ord(t) < tSolveFirst + mSettings(m, 't_reserveLength') // Unit is either providing
                                                        and sum(restype, nuRescapable(restype, 'down', node, unit)) // downward reserves
                                                        ]
                                                    // NOTE!!! Could be better to form a gnuft_reserves subset?
                                                    or [ // the unit has an online variable
                                                        uft_online(unit, f, t)
                                                        and [
                                                            (unit_minLoad(unit) and p_gnu(grid, node, unit, 'unitSizeGen')) // generators with a min. load
                                                            or p_gnu(grid, node, unit, 'maxCons') // or consuming units with an online variable
                                                            ]
                                                        ] // END or
                                                    or [ // consuming units with investment possibility
                                                        gnu_input(grid, node, unit)
                                                        and [unit_investLP(unit) or unit_investMIP(unit)]
                                                        ]
                                                    } ..
    // Energy generation/consumption
    + v_gen(grid, node, unit, f, t)

    // Considering output constraints (e.g. cV line)
408
409
410
    + sum(gngnu_constrainedOutputRatio(grid, node, grid_output, node_, unit),
        + p_gnu(grid_output, node_, unit, 'cV')
            * v_gen(grid_output, node_, unit, f, t)
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417
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420
421
        ) // END sum(gngnu_constrainedOutputRatio)

    // Downward reserve participation
    - sum(nuRescapable(restype, 'down', node, unit)${ord(t) < tSolveFirst + mSettings(m, 't_reserveLength')},
        + v_reserve(restype, 'down', node, unit, f+df_nReserves(node, restype, f, t), t) // (v_reserve can be used only if the unit is capable of providing a particular reserve)
        ) // END sum(nuRescapable)

    =G= // Must be greater than minimum load or maximum consumption  (units with min-load and both generation and consumption are not allowed)

    // Generation units, greater than minload
    + p_gnu(grid, node, unit, 'unitSizeGen')
422
        * sum(suft(effGroup, unit, f, t), // Uses the minimum 'lb' for the current efficiency approximation
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            + p_effGroupUnit(effGroup, unit, 'lb')${not ts_effGroupUnit(effGroup, unit, 'lb', f, t)}
            + ts_effGroupUnit(effGroup, unit, 'lb', f, t)
            ) // END sum(effGroup)
        * [ // Online variables should only be generated for units with restrictions
427
428
            + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)} // LP online variant
            + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)} // MIP online variant
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430
            ] // END v_online

431
    // Units that are in the run-up phase need to keep up with the run-up ramp rate (contained in p_ut_runUp)
432
    + p_gnu(grid, node, unit, 'unitSizeGen')
433
434
        * sum(t_activeNoReset(t_)${ ord(t_) > ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
                                    and ord(t_) <= ord(t) and uft_online(unit, f, t)},
435
            + sum(unitStarttype(unit, starttype),
436
                + v_startup(unit, starttype, f+df_central(f,t), t_)
437
                    * sum(t_full(t__)${ord(t__) = p_u_runUpTimeIntervalsCeil(unit) - ord(t) - dt_next(t) + 1 + ord(t_)}, // last step in the interval
438
439
                        + p_ut_runUp(unit, t__)
*                            * 1 // test values [0,1] to provide some flexibility
440
441
442
                        ) // END sum(t__)
                ) // END sum(unitStarttype)
            )$p_u_runUpTimeIntervals(unit)  // END sum(t_)
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444
    // Units that are in the last time interval of the run-up phase are limited by the minimum load (contained in p_ut_runUp(unit, 't00000'))
    + p_gnu(grid, node, unit, 'unitSizeGen')
445
446
        * sum(t_activeNoReset(t_)${ ord(t_) = ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
                                    and uft_online(unit, f, t)},
447
448
            + sum(unitStarttype(unit, starttype),
                + v_startup(unit, starttype, f+df_central(f,t), t_)
449
                    * sum(t_full(t__)${ord(t__) = 1}, p_ut_runUp(unit, t__))
450
451
                ) // END sum(unitStarttype)
            )$p_u_runUpTimeIntervals(unit)  // END sum(t_)
452

453
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457
    // Consuming units, greater than maxCons
    // Available capacity restrictions
    - p_unit(unit, 'availability')
        * [
            // Capacity factors for flow units
458
            + sum(flowUnit(flow, unit),
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463
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465
466
467
                + ts_cf_(flow, node, f, t)
                ) // END sum(flow)
            + 1${not unit_flow(unit)}
            ] // END * p_unit(availability)
        * [
            // Online capacity restriction
            + p_gnu(grid, node, unit, 'maxCons')${not uft_online(unit, f, t)} // Use initial maximum if no online variables
            + p_gnu(grid, node, unit, 'unitSizeCons')
                * [
468
                    // Capacity online
469
470
                    + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
                    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
471
472
473
474
475
476
477
478

                    // Investments to additional non-online capacity
                    + sum(t_invest(t_)${    ord(t_)<=ord(t)
                                            and not uft_online(unit, f, t)
                                            },
                        + v_invest_LP(unit, t_)${unit_investLP(unit)} // NOTE! v_invest_LP also for consuming units is positive
                        + v_invest_MIP(unit, t_)${unit_investMIP(unit)} // NOTE! v_invest_MIP also for consuming units is positive
                        ) // END sum(t_invest)
479
480
                    ] // END * p_gnu(unitSizeCons)
            ] // END * p_unit(availability)
481
;
482

483
484
// Not sure if this is needed because we already have q_maxDownward
q_noReserveInRunUp(m, gnuft(grid, node, unit, f, t))$[   ord(t) < tSolveFirst + mSettings(m, 't_reserveLength') // Unit is both providing
485
                                                    and sum(restype, nuRescapable(restype, 'up', node, unit)) // upward reserves
486
                                                    and p_u_runUpTimeIntervals(unit)   // and unit has run up constraint
487
488
489
490
                                                    ]..
    v_gen(grid, node, unit, f, t)
    =G=
    + p_gnu(grid, node, unit, 'unitSizeGen')
491
        * sum(t_activeNoReset(t_)$(ord(t_) > ord(t) + dt_toStartup(unit, t) and ord(t_) <= ord(t) and uft_online(unit, f, t_)),
492
            + sum(unitStarttype(unit, starttype),
493
                + v_startup(unit, starttype, f+df_central(f,t), t_) * sum(t__${ord(t__) = ord(t) - ord(t_) + 1}, p_ut_runUp(unit, t__))  //t+dtt(t,t_)
494
495
496
497
498
499
500
501
502
503
504
            )
          )$p_u_runUpTimeIntervals(unit)

$ontext
    p_nuReserves(node, unit, resType, 'up')
      * (
          + p_unit(unit, 'unitCount')
          + sum(t_invest(t_)${ ord(t_)<=ord(t) },
               + v_invest_LP(unit, t_)${unit_investLP(unit)} // NOTE! v_invest_LP also for consuming units is positive
               + v_invest_MIP(unit, t_)${unit_investMIP(unit)} // NOTE! v_invest_MIP also for consuming units is positive
            ) // END sum(t_invest)
505
          - sum(t_$(ord(t_) >= ord(t) + dt_toStartup(unit, t) and ord(t_) < ord(t) and uft_online(unit, f, t_)),
506
              + sum(unitStarttype(unit, starttype),
507
                  + v_startup(unit, starttype, f+df_central(f,t), t_)
508
509
510
511
512
513
                )
            )
        ) * p_gnu(grid, node, unit, 'unitSizeGen')
$offtext
;

514
515
516
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521
522
523
524
525
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533
534
535
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540
541
542
543
544
545
546
547
548
* --- Maximum Upwards Capacity ------------------------------------------------

q_maxUpward(m, gnuft(grid, node, unit, f, t))${ [   ord(t) < tSolveFirst + mSettings(m, 't_reserveLength') // Unit is either providing
                                                    and sum(restype, nuRescapable(restype, 'up', node, unit)) // upward reserves
                                                    ]
                                                or [
                                                    uft_online(unit, f, t) // or the unit has an online variable
                                                        and [
                                                            [unit_minLoad(unit) and p_gnu(grid, node, unit, 'unitSizeCons')] // consuming units with min_load
                                                            or [p_gnu(grid, node, unit, 'maxGen')]                          // generators with an online variable
                                                            ]
                                                    ]
                                                or [
                                                    gnu_output(grid, node, unit) // generators with investment possibility
                                                    and (unit_investLP(unit) or unit_investMIP(unit))
                                                    ]
                                                }..
    // Energy generation/consumption
    + v_gen(grid, node, unit, f, t)

    // Considering output constraints (e.g. cV line)
    + sum(gngnu_constrainedOutputRatio(grid, node, grid_output, node_, unit),
        + p_gnu(grid_output, node_, unit, 'cV')
            * v_gen(grid_output, node_, unit, f, t)
        ) // END sum(gngnu_constrainedOutputRatio)

    // Upwards reserve participation
    + sum(nuRescapable(restype, 'up', node, unit)${ord(t) < tSolveFirst + mSettings(m, 't_reserveLength')},
        + v_reserve(restype, 'up', node, unit, f+df_nReserves(node, restype, f, t), t)
        ) // END sum(nuRescapable)

    =L= // must be less than available/online capacity

    // Consuming units
    + p_gnu(grid, node, unit, 'unitSizeCons')
549
        * sum(suft(effGroup, unit, f, t), // Uses the minimum 'lb' for the current efficiency approximation
550
551
552
553
            + p_effGroupUnit(effGroup, unit, 'lb')${not ts_effGroupUnit(effGroup, unit, 'lb', f, t)}
            + ts_effGroupUnit(effGroup, unit, 'lb', f, t)
            ) // END sum(effGroup)
        * [
554
555
            + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)} // Consuming units are restricted by their min. load (consuming is negative)
            + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)} // Consuming units are restricted by their min. load (consuming is negative)
556
557
558
559
560
561
562
            ] // END * p_gnu(unitSizeCons)

    // Generation units
    // Available capacity restrictions
    + p_unit(unit, 'availability') // Generation units are restricted by their (available) capacity
        * [
            // Capacity factor for flow units
563
            + sum(flowUnit(flow, unit),
564
565
566
567
568
569
570
571
572
                + ts_cf_(flow, node, f, t)
                ) // END sum(flow)
            + 1${not unit_flow(unit)}
            ] // END * p_unit(availability)
        * [
            // Online capacity restriction
            + p_gnu(grid, node, unit, 'maxGen')${not uft_online(unit, f, t)} // Use initial maxGen if no online variables
            + p_gnu(grid, node, unit, 'unitSizeGen')
                * [
573
                    // Capacity online
574
575
                    + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f ,t)}
                    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
576
577
578
579
580
581
582
583

                    // Investments to non-online capacity
                    + sum(t_invest(t_)${    ord(t_)<=ord(t)
                                            and not uft_online(unit, f ,t)
                                            },
                        + v_invest_LP(unit, t_)${unit_investLP(unit)}
                        + v_invest_MIP(unit, t_)${unit_investMIP(unit)}
                        ) // END sum(t_invest)
584
585
                    ] // END * p_gnu(unitSizeGen)
            ] // END * p_unit(availability)
586

587
    // Units that are in the run-up phase need to keep up with the run-up ramp rate (contained in p_ut_runUp)
588
    + p_gnu(grid, node, unit, 'unitSizeGen')
589
590
        * sum(t_activeNoReset(t_)${ ord(t_) > ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
                                    and ord(t_) <= ord(t) and uft_online(unit, f, t)},
591
            + sum(unitStarttype(unit, starttype),
592
                + v_startup(unit, starttype, f+df_central(f,t), t_)
593
                    * sum(t_full(t__)${ord(t__) = p_u_runUpTimeIntervalsCeil(unit) - ord(t) - dt_next(t) + 1 + ord(t_)}, // last step in the interval
594
595
596
597
                        + p_ut_runUp(unit, t__)
                      ) // END sum(t__)
              ) // END sum(unitStarttype)
          )$p_u_runUpTimeIntervals(unit) // END sum(t_)
598
    // Units that are in the last time interval of the run-up phase are limited by the p_u_maxOutputInLastRunUpInterval
599
    + p_gnu(grid, node, unit, 'unitSizeGen')
600
        * sum(t_activeNoReset(t_)${ ord(t_) = ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
601
                   and uft_online(unit, f, t)},
602
            + sum(unitStarttype(unit, starttype),
603
                + v_startup(unit, starttype, f+df_central(f,t), t_) * p_u_maxOutputInLastRunUpInterval(unit)
604
605
              ) // END sum(unitStarttype)
          )$p_u_runUpTimeIntervals(unit) // END sum(t_)
606
;
607
608
609

* --- Unit Startup and Shutdown -----------------------------------------------

610
q_startshut(m, uft_online(unit, f, t))${ ord(t) + dt(t) > mSettings(m, 't_start') } ..
611
    // Units currently online
612
613
    + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
614
615

    // Units previously online
616
617
    - v_online_LP(unit, f+df_central(f,t+dt(t)), t+dt(t))${ uft_onlineLP(unit, f, t) } // This reaches to tFirstSolve when dt = -1
    - v_online_MIP(unit, f+df_central(f,t+dt(t)), t+dt(t))${ uft_onlineMIP(unit, f, t) }
618
619

    // Unit online history (solve initial value), required because uft_online doesn't extend to before active modelling
620
621
622
*    - r_online(unit, f+df_central(f,t+dt(t)), t+dt(t))${    not uft_onlineLP(unit, f+df(f,t+dt(t)), t+dt(t))
*                                                    and not uft_onlineMIP(unit, f+df(f,t+dt(t)), t+dt(t))
*                                                    }
623

624
625
    =E=

626
    // Unit startup and shutdown
627
    + sum(unitStarttype(unit, starttype),
628
        + v_startup(unit, starttype, f+df_central(f,t+dt_toStartup(unit,t)), t+dt_toStartup(unit, t))
629
        ) // END sum(starttype)
630
    - v_shutdown(unit, f+df_central(f,t), t)
631
;
632

633

634
*--- Startup Type -------------------------------------------------------------
635
// !!! NOTE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
636
637
638
// This formulation doesn't work as intended when unitCount > 1, as one recent
// shutdown allows for multiple hot/warm startups on subsequent time steps.
// Pending changes.
639

640
q_startuptype(m, starttypeConstrained(starttype), uft_online(unit, f, t))${ unitStarttype(unit, starttype) } ..
641
642

    // Startup type
643
    + v_startup(unit, starttype, f+df_central(f,t), t)
644
*Experimental    + sum[ft(f_, t_)${uft_online(unit, f_, t_) and ord(t_) < ord(t)}, v_startup(unit, starttype, f+df_central(f,t_+dt_toStartup(unit,t_)), t_+dt_toStartup(unit, t_))]
645
646
647
648

    =L=

    // Subunit shutdowns within special startup timeframe
649
    + sum(counter${dt_starttypeUnitCounter(starttype, unit, counter)},
650
        + v_shutdown(unit, f+df_central(f,t+(dt_starttypeUnitCounter(starttype, unit, counter)+1)), t+(dt_starttypeUnitCounter(starttype, unit, counter)+1))
651
    ) // END sum(counter)
652
;
653

654

655
656
*--- Online Limits with Startup Type Constraints and Investments --------------

657
q_onlineLimit(m, uft_online(unit, f, t))${  p_unit(unit, 'minShutdownHours')
658
                                            or p_u_runUpTimeIntervals(unit)
659
660
661
662
                                            or unit_investLP(unit)
                                            or unit_investMIP(unit)
                                            } ..
    // Online variables
663
664
    + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f ,t)}
665
666
667
668
669
670

    =L=

    // Number of existing units
    + p_unit(unit, 'unitCount')

671
    // Number of units unable to become online due to restrictions
672
    - sum(counter${dt_downtimeUnitCounter(unit, counter)},
673
        + v_shutdown(unit, f+df_central(f,t+(dt_downtimeUnitCounter(unit, counter) + 1)), t+(dt_downtimeUnitCounter(unit, counter) + 1))
674
    ) // END sum(counter)
675
676
677

    // Investments into units
    + sum(t_invest(t_)${ord(t_)<=ord(t)},
678
679
        + v_invest_LP(unit, t_)${unit_investLP(unit)}
        + v_invest_MIP(unit, t_)${unit_investMIP(unit)}
680
681
682
        ) // END sum(t_invest)
;

683
684
685
686
*--- Both q_offlineAfterShutdown and q_onlineOnStartup work when there is only one unit.
*    These equations prohibit single units turning on and off at the same time step.
*    Unfortunately there seems to be no way to prohibit this when unit count is > 1.
*    (it shouldn't be worthwhile anyway if there is a startup cost, but it can fall within the solution gap).
687
688
689
690
691
692
693
694
695
q_onlineOnStartUp(uft_online(unit, f, t))${sum(starttype, unitStarttype(unit, starttype))}..

    // Units currently online
    + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}

    =G=

    + sum(unitStarttype(unit, starttype),
696
        + v_startup(unit, starttype, f+df_central(f,t+dt_toStartup(unit, t)), t+dt_toStartup(unit, t))  //dt_toStartup displaces the time step to the one where the unit would be started up in order to reach online at t
697
698
699
700
701
      ) // END sum(starttype)
;

q_offlineAfterShutdown(uft_online(unit, f, t))${sum(starttype, unitStarttype(unit, starttype))}..

702
703
704
705
706
    // Number of existing units
    + p_unit(unit, 'unitCount')

    // Investments into units
    + sum(t_invest(t_)${ord(t_)<=ord(t)},
707
708
        + v_invest_LP(unit, t_)${unit_investLP(unit)}
        + v_invest_MIP(unit, t_)${unit_investMIP(unit)}
709
710
        ) // END sum(t_invest)

711
712
713
714
715
716
    // Units currently online
    - v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
    - v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}

    =G=

717
    + v_shutdown(unit, f+df_central(f,t), t)
718
719
;

720
721
*--- Minimum Unit Uptime ------------------------------------------------------

722
q_onlineMinUptime(m, uft_online(unit, f, t))${  p_unit(unit, 'minOperationHours')
723
724
725
                                                } ..

    // Units currently online
726
727
    + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
728
729
730
731

    =G=

    // Units that have minimum operation time requirements active
732
    + sum(counter${dt_uptimeUnitCounter(unit, counter)},
733
        + sum(unitStarttype(unit, starttype),
734
            + v_startup(unit, starttype, f+df_central(f,t+(dt_uptimeUnitCounter(unit, counter)+dt_toStartup(unit, t) + 1)), t+(dt_uptimeUnitCounter(unit, counter)+dt_toStartup(unit, t) + 1))
735
            ) // END sum(starttype)
736
    ) // END sum(counter)
737
738
;

739
* --- Ramp Constraints --------------------------------------------------------
740
q_genRamp(m, gn(grid, node), s, uft(unit, f, t))${  gnuft_ramp(grid, node, unit, f, t)
741
                                                    and ord(t) > msStart(m, s) + 1
742
                                                    and msft(m, s, f, t)
743
744
                                                    } ..

745
    + v_genRamp(grid, node, unit, f, t)
746
        * p_stepLength(m, f, t)
747
748
    =E=
    // Change in generation over the time step
749
    + v_gen(grid, node, unit, f, t)
750
    - v_gen(grid, node, unit, f+df(f,t+dt(t)), t+dt(t))
751
;
752

753
* --- Ramp Up Limits ----------------------------------------------------------
754
755
756
757
q_rampUpLimit(m, gn(grid, node), s, uft(unit, f, t))${  gnuft_ramp(grid, node, unit, f, t)
                                                        and ord(t) > msStart(m, s) + 1
                                                        and msft(m, s, f, t)
                                                        and p_gnu(grid, node, unit, 'maxRampUp')
758
                                                   } ..
759
  + v_genRamp(grid, node, unit, f, t)
760
761
762
  + sum(nuRescapable(restype, 'up', node, unit)${ord(t) < tSolveFirst + mSettings(m, 't_reserveLength')},
      + v_reserve(restype, 'up', node, unit, f+df_nReserves(node, restype, f, t), t) // (v_reserve can be used only if the unit is capable of providing a particular reserve)
      ) // END sum(nuRescapable)
763
  =L=
764
    // Ramping capability of units without an online variable
765
  + (
766
      + ( p_gnu(grid, node, unit, 'maxGen') + p_gnu(grid, node, unit, 'maxCons') )${not uft_online(unit, f, t)}
767
      + sum(t_invest(t_)${ ord(t_)<=ord(t) },
768
          + v_invest_LP(unit, t_)${not uft_onlineLP(unit, f, t) and unit_investLP(unit)}
769
              * p_gnu(grid, node, unit, 'unitSizeTot')
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          + v_invest_MIP(unit, t_)${not uft_onlineMIP(unit, f, t) and unit_investMIP(unit)}
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              * p_gnu(grid, node, unit, 'unitSizeTot')
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        )
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    )
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      * p_gnu(grid, node, unit, 'maxRampUp')
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      * 60   // Unit conversion from [p.u./min] to [p.u./h]
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    // Ramping capability of units with an online variable
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  + (
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      + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
      + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
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    )
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      * p_gnu(grid, node, unit, 'unitSizeTot')
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      * p_gnu(grid, node, unit, 'maxRampUp')
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      * 60   // Unit conversion from [p.u./min] to [p.u./h]
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    // Units that are in the run-up phase need to keep up with the run-up ramp rate (contained in p_ut_runUp)
  + p_gnu(grid, node, unit, 'unitSizeGen')
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      * sum(t_activeNoReset(t_)${   ord(t_) > ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
                                    and ord(t_) <= ord(t) and uft_online(unit, f, t)},
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          + sum(unitStarttype(unit, starttype),
              + v_startup(unit, starttype, f+df_central(f,t), t_)
                  * p_unit(unit, 'rampSpeedToMinLoad')
                  * 60   // Unit conversion from [p.u./min] to [p.u./h]
            ) // END sum(unitStarttype)
        )$p_u_runUpTimeIntervals(unit) // END sum(t_)
    // Units that are in the last time interval of the run-up phase are limited by the p_u_maxOutputInLastRunUpInterval
  + p_gnu(grid, node, unit, 'unitSizeGen')
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      * sum(t_activeNoReset(t_)${   ord(t_) = ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
                                    and uft_online(unit, f, t)},
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          + sum(unitStarttype(unit, starttype),
              + v_startup(unit, starttype, f+df_central(f,t), t_)
                  * max(p_unit(unit, 'rampSpeedToMinLoad'), p_gnu(grid, node, unit, 'maxRampUp')) // could also be weighted average from 'maxRampUp' and 'rampSpeedToMinLoad'
                  * 60   // Unit conversion from [p.u./min] to [p.u./h]
            ) // END sum(unitStarttype)
        )$p_u_runUpTimeIntervals(unit) // END sum(t_)
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    // Shutdown of consumption units from full load
  + v_shutdown(unit, f+df_central(f,t), t)${uft_online(unit, f, t) and gnu_input(grid, node, unit)}
      * p_gnu(grid, node, unit, 'unitSizeTot')
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// Note: This constraint does not limit ramping properly for example if online subunits are
// producing at full capacity (= not possible to ramp up) and more subunits are started up.
// Take this into account in q_maxUpward or in another equation?:
// v_gen =L= (v_online(t-1) - v_shutdown(t-1)) * unitSize + v_startup(t-1) * unitSize * minLoad
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;
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* --- Ramp Down Limits --------------------------------------------------------
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q_rampDownLimit(gn(grid, node), m, s, uft(unit, f, t))${    gnuft_ramp(grid, node, unit, f, t)
                                                            and ord(t) > msStart(m, s) + 1
                                                            and msft(m, s, f, t)
                                                            and p_gnu(grid, node, unit, 'maxRampDown')
                                                            } ..
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  + v_genRamp(grid, node, unit, f, t)
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  - sum(nuRescapable(restype, 'down', node, unit)${ord(t) < tSolveFirst + mSettings(m, 't_reserveLength')},
      + v_reserve(restype, 'down', node, unit, f+df_nReserves(node, restype, f, t), t) // (v_reserve can be used only if the unit is capable of providing a particular reserve)
      ) // END sum(nuRescapable)
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  =G=
    // Ramping capability of units without online variable
  - (
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      + ( p_gnu(grid, node, unit, 'maxGen') + p_gnu(grid, node, unit, 'maxCons') )${not uft_online(unit, f, t)}
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      + sum(t_invest(t_)${ ord(t_)<=ord(t) },
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          + v_invest_LP(unit, t_)${not uft_onlineLP(unit, f, t) and unit_investLP(unit)}
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              * p_gnu(grid, node, unit, 'unitSizeTot')
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          + v_invest_MIP(unit, t_)${not uft_onlineMIP(unit, f, t) and unit_investMIP(unit)}
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              * p_gnu(grid, node, unit, 'unitSizeTot')
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        )
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    )
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      * p_gnu(grid, node, unit, 'maxRampDown')
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      * 60   // Unit conversion from [p.u./min] to [p.u./h]
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    // Ramping capability of units that are online
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  - (
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      + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
      + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
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    )
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      * p_gnu(grid, node, unit, 'unitSizeTot')
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      * p_gnu(grid, node, unit, 'maxRampDown')
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      * 60   // Unit conversion from [p.u./min] to [p.u./h]
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    // Shutdown of generation units from full load
  - v_shutdown(unit, f+df_central(f,t), t)${uft_online(unit, f, t) and gnu_output(grid, node, unit)}
      * p_gnu(grid, node, unit, 'unitSizeTot')
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;


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* --- Fixed Output Ratio ------------------------------------------------------

q_outputRatioFixed(gngnu_fixedOutputRatio(grid, node, grid_, node_, unit), ft(f, t))${  uft(unit, f, t)
                                                                                        } ..

    // Generation in grid
    + v_gen(grid, node, unit, f, t)
        / p_gnu(grid, node, unit, 'cB')

    =E=

    // Generation in grid_
    + v_gen(grid_, node_, unit, f, t)
        / p_gnu(grid_, node_, unit, 'cB')
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;
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* --- Constrained Output Ratio ------------------------------------------------

q_outputRatioConstrained(gngnu_constrainedOutputRatio(grid, node, grid_, node_, unit), ft(f, t))${  uft(unit, f, t)
                                                                                                    } ..

    // Generation in grid
    + v_gen(grid, node, unit, f, t)
        / p_gnu(grid, node, unit, 'cB')

    =G=

    // Generation in grid_
    + v_gen(grid_, node_, unit, f, t)
        / p_gnu(grid_, node_, unit, 'cB')
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;
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* --- Direct Input-Output Conversion ------------------------------------------

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q_conversionDirectInputOutput(suft(effDirect(effGroup), unit, f, t)) ..
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    // Sum over endogenous energy inputs
    - sum(gnu_input(grid, node, unit),
        + v_gen(grid, node, unit, f, t)
        ) // END sum(gnu_input)

    // Sum over fuel energy inputs
    + sum(uFuel(unit, 'main', fuel),
        + v_fuelUse(fuel, unit, f, t)
        ) // END sum(uFuel)

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    // Is main fuel used also in the run-up phase when having directOnMIP or directOnLP?

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    =E=

    // Sum over energy outputs
    + sum(gnu_output(grid, node, unit),
        + v_gen(grid, node, unit, f, t)
            * [ // Heat rate
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                + p_effUnit(effGroup, unit, effGroup, 'slope')${ not ts_effUnit(effGroup, unit, effGroup, 'slope', f, t) }
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                + ts_effUnit(effGroup, unit, effGroup, 'slope', f, t)
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                ] // END * v_gen
        ) // END sum(gnu_output)

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    // Consumption of keeping units online (no-load fuel use)
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    + sum(gnu_output(grid, node, unit),
        + p_gnu(grid, node, unit, 'unitSizeGen')
        ) // END sum(gnu_output)
        * [
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            + v_online_LP(unit, f+df_central(f,t), t)${uft_onlineLP(unit, f, t)}
            + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
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            ] // END * sum(gnu_output)
        * [
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            + p_effGroupUnit(effGroup, unit, 'section')${not ts_effUnit(effGroup, unit, effDirect, 'section', f, t)}
            + ts_effUnit(effGroup, unit, effGroup, 'section', f, t)
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            ] // END * sum(gnu_output)
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* --- SOS2 Efficiency Approximation -------------------------------------------

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q_conversionSOS2InputIntermediate(suft(effLambda(effGroup), unit, f, t)) ..

    // Sum over endogenous energy inputs
    - sum(gnu_input(grid, node, unit),
        + v_gen(grid, node, unit, f, t)
        ) // END sum(gnu_input)

    // Sum over fuel energy inputs
    + sum(uFuel(unit, 'main', fuel),
        + v_fuelUse(fuel, unit, f, t)
        ) // END sum(uFuel)

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    =G=
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    // Sum over the endogenous outputs of the unit
    + sum(gnu_output(grid, node, unit), p_gnu(grid, node, unit, 'unitSizeGen'))
        * [
            // Consumption of generation
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            + sum(effGroupSelectorUnit(effGroup, unit, effSelector),
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                + v_sos2(unit, f, t, effSelector)
                    * [ // Operation points convert the v_sos2 variables into share of capacity used for generation
                        + p_effUnit(effGroup, unit, effSelector, 'op')${not ts_effUnit(effGroup, unit, effSelector, 'op', f, t)}
                        + ts_effUnit(effGroup, unit, effSelector, 'op', f, t)
                        ] // END * v_sos2
                    * [ // Heat rate
                        + p_effUnit(effGroup, unit, effSelector, 'slope')${not ts_effUnit(effGroup, unit, effSelector, 'slope', f, t)}
                        + ts_effUnit(effGroup, unit, effSelector, 'slope', f, t)
                        ] // END * v_sos2
                ) // END sum(effSelector)

            // Consumption of keeping units online
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            + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
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                * p_effGroupUnit(effGroup, unit, 'section')
            ] // END * sum(gnu_output)
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;
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* --- SOS 2 Efficiency Approximation Online Variables -------------------------

q_conversionSOS2Constraint(suft(effLambda(effGroup), unit, f, t)) ..

    // Total value of the v_sos2 equals the number of online units
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    + sum(effGroupSelectorUnit(effGroup, unit, effSelector),
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        + v_sos2(unit, f, t, effSelector)
        ) // END sum(effSelector)

    =E=

    // Number of units online
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    + v_online_MIP(unit, f+df_central(f,t), t)${uft_onlineMIP(unit, f, t)}
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;
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* --- SOS 2 Efficiency Approximation Output Generation ------------------------

q_conversionSOS2IntermediateOutput(suft(effLambda(effGroup), unit, f, t)) ..

    // Endogenous energy output
    + sum(gnu_output(grid, node, unit),
        + p_gnu(grid, node, unit, 'unitSizeGen')
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      ) // END sum(gnu_output)
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        * sum(effGroupSelectorUnit(effGroup, unit, effSelector),
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            + v_sos2(unit, f, t, effSelector)
            * [ // Operation points convert v_sos2 into share of capacity used for generation
                + p_effUnit(effGroup, unit, effSelector, 'op')${not ts_effUnit(effGroup, unit, effSelector, 'op', f, t)}
                + ts_effUnit(effGroup, unit, effSelector, 'op', f, t)
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              ] // END * v_sos2
          ) // END sum(effSelector)
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    // Units that are in the run-up phase need to keep up with the run-up ramp rate (contained in p_ut_runUp)
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    + sum(gnu_output(grid, node, unit)$p_u_runUpTimeIntervals(unit),
        + p_gnu(grid, node, unit, 'unitSizeGen')
      ) // END sum(gnu_output)
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        * sum(t_activeNoReset(t_)${ ord(t_) > ord(t) + dt_next(t) + dt_toStartup(unit, t + dt_next(t))
                                    and ord(t_) <= ord(t) and uft_online(unit, f, t)
                                    },
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            + sum(unitStarttype(unit, starttype),
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                + v_startup(unit, starttype, f+df_central(f,t), t_)