The capacity expansion problem is formulated in the simulator as a Mixed-Integer Linear Program (MILP or MIP for short). To illustrate the very core of this formulation, we develop the following simplified formulation.
NOTE: The following description is illustrative only, and shows only the core of the LT Plan MIP formulation.
Definitions
Let us define the following variables:
| Variable | Description | Type |
|---|---|---|
| GenBuild(g,y) | Number of generating units build in year y for Generator g | integer |
| GenLoad(g,t) | Dispatch level of generating unit g in period t | continuous |
| USEt | Unserved energy in dispatch period t | continuous |
| CapShorty | Capacity shortage in year y | continuous |
We also define the following parameters:
| Element | Description | Unit |
|---|---|---|
| D | Discount rate. We then derive DFy = 1/(1 + D)y which is the discount factor applied to year, and DFt which is the discount factor applied to dispatch period t | |
| Lt | Duration of dispatch period t | Hours |
| BuildCostg | Overnight build cost of generator g | $ |
| MaxUnitsBuilt(g,y) | Maximum number of units of generator g allowed to be built by the end of year y | |
| PMAXg | Maximum generating capacity of each unit of generator g | MW |
| Unitsg | Number of installed generating units of generator g | |
| VoLL | Value of lost load (energy shortage price) | $/MWh |
| SRMCg | Short-run marginal cost of generator g which is composed of Heat Rate × Fuel Price + VO&M Charge | $/MWh |
| FOMChargeg | Fixed operations and maintenance charge of generator g | $ |
| Loadt | Average power demand in dispatch period t | MW |
| PeakLoady | System peak power demand in year y | MW |
| ReserveMarginy | Margin required over maximum power demand in year y | MW |
| CapShortPrice | Capacity shortage price | $/MW |
Note this illustrative formulation includes only build decisions, but can easily be extended to include retirements.
The objective function of LT Plan seeks to minimize the net present value of build costs plus fixed operations and maintenance costs plus production costs. The core formulation for LT Plan is thus:
Minimize
∑ ( y ) ∑ ( g ) DFy × ( BuildCostg × GenBuild(g,y) )
+∑ ( y ) DFy X [ FOMChargeg × 1000 × PMAXg ( Unitsg + ∑ i≤y GenBuild g,i )]
+∑ ( t ) DFt∈y × Lt × [ VoLL × USE t + ∑ g (SRMCg × GenLoadg,t )]
subject to
Equation 1: Energy Balance
∑ ( g ) GenLoad(g,y)
+ USEt = Demandt ∀t
Equation 2: Feasible Energy Dispatch
GenLoad(g,t) ≤ PMAX
( Unitsg +
∑ i≤y GenBuild g,i
)
Equation 3: Feasible Builds
∑ i≤y GenBuild
g,i ≤ MaxUnitsBuiltg,y
Equation 4: Integrality
GenBuild(g,y) integer
Note that the LT Plan formulation in its minimal form does not include constraints on capacity margin. Instead the natural trade-off between energy shortage and build costs will ensure that capacity is built if it is economic and that the energy price (dual variable associated with the energy balance constraints) will exactly compensate the marginal build for its production and build costs. The resulting capacity reserve margin could take any value (including negative) and the amount of unserved energy could be any value up to the amount of load i.e. this 'natural' solution may or may not meet acceptable reliability standards.
In reality, systems require a certain minimum capacity reserve, which is commonly calculated based on reliability metrics such as LOLP or EENS. These values are not easily modelled directly in the MIP formulation and thus must be converted to a reserve margin (percentage of peak load) or absolute megawatt capacity requirement instead. Thus the additional equation would appear in the formulation:
Equation 5: Capacity Adequacy
∑ ( g ) PMAXg
( Unitsg
+ ∑ i≤y GenBuildi
) + CapShorty
≥ PeakLoady + ReserveMarginy
∀y
The effect of this constraint is to ensure that new builds occur not just for economic reasons but also to meet the capacity reserve margin: and this constraint may also defer economic retirements.
If the constraint is binding at the optimal solution its shadow price represents a Capacity Price, and the energy price will not, on its own, compensate the marginal build for its build and production costs i.e. compensation comes from a combination of energy and Capacity Payments.
However, the pricing outcome is more complicated than this due to the integer nature of the decision variables involved. In most cases in fact the shadow price on this capacity constraint will be zero, even though it has changed the solution-an effect referred to as a 'duality gap'. This is because capacity is added in 'lumps' rather than on a continuous basis, thus in most cases the constraint is slack at the integer optimal solution-see Revenue Adequacy.
However, LT Plan automatically computes a pseudo 'capacity shadow price' in the "Integer Optimal" case based on the loss incurred by the marginal new build in each capacity Region/Zone.
It is assumed that neither forced outages nor maintenance outages affect capacity, but that both forced and maintenance outages subtract from available energy.
Generator Forced Outage Rate (FOR) and Maintenance Rate (MOR) are included in the LT Plan formulation. FOR and MOR subtract from the energy contribution of generators, thus Equation 2 changes to:
Equation 6: Energy Dispatch Feasibility Accounting for
Outage Rates
GenLoadg,t ≤ ( 1 - MORg
× MFt - FORg ) × PMAXg
× ( Unitsg
+ ∑ ( i ≤ y
) ( GenBuildi ) )
∀ g,t
where:
MFt is the Region Maintenance Factor in period t
When considering the capacity available to meet a capacity margin constraint forced outage and maintenance is ignored in summing generation capacity, since the input reserve margin should already account for forced outages.
The LT Plan attribute Use Effective Load Approach (ELA) provides another means for accounting for forced outage rates. By default the ELA is "off" and the capacity constraints is:
Equation 7: Capacity Constraint
∑ ( g ) PMAXg ×
( Unitsg + ∑ ( i ≤ y
) ( GenBuild i ))
+ CapShorty ≥ PeakLoady +
ReserveMarginy ∀y
If the ELA is enabled however, the 'raw' generation capacity is used in this equation, but the load is modified, thus the input minimum reserve margin could be zero. The basic premise behind the Effective Load Approach is to take the load and appropriately modify it with respect to the convolved capacity outages. It transposes the convolved generators forced outage rates into the load, modifying the load to account for the capacity outages.
Equation 8: Generation Capacity with ELA
GenLoadg,t ≤ ( 1 - MORg
× MFt - FORg ) × PMAXg
× ( Unitsg
+ ∑ ( i ≤ y
) ( GenBuildi ) )
∀ g,t
The Generator Build Cost property represents the (per kW) overnight cost of building a new unit, i.e. the all-in capital cost as per commissioning date. The above formulation shows this cost being incurred in the year the plant is built. This 'lumpy' capital cost approach is really only suitable if the planning horizon is long enough to fully amortize the build cost e.g. at least the Economic Life of the plant. If the horizon is not long enough to cover the economic life of the last new build then the 'lumpy' cost model is incorrect.
The approach used for projects whose life extends past the end of the planning horizon, is to annualize the build costs i.e. convert them to an equivalent annual charge which is applied in the year of build and every subsequent year across the economic life of the generator. Note that Economic Life can be different to the Technical Life. The former defines the period over which the aggregate annualized charge should equal the build cost, and the latter is the physical lifetime of the plant.
We transform the 'lumpy' build cost input into an annualised charge suitable for the LT Plan formulation as follows:
Equation 9: Annuity Calculation
Annuity{g,y} = Build
Cost{g,y} × 1000 × Max
Capacityg,y × WACCg,y
/ ( 1 - [
1 / ( 1 + WACCg
) ] Economic
Life )
The build cost coefficient in the objective function (i.e. BuildCostg) is thus replaced by the sum of the discounted annual charges, starting from the given year y until the end of the economic life of the unit. The discount rate used is the system discount rate, whereas the Generator WACC (weighted average cost of capital) will be project-specific. If no WACC rate is set, the discount rate is used.
This calculation applies when no depreciation method is selected. Straight-line and declining balance methods are also available. To view the formulation of these methods, see Depreciation Benefits.
It is important, in the LT Plan formulation that the model does not inappropriately consider the end of the planning horizon to also be the 'end of time'. In the above basic formulation we have implicitly assumed that LT Plan has an infinite horizon. In practice, the model will have a finite horizon. We therefore have to take account of so-called 'end year effects'.
Without some method for dealing with end-year effects, the model will in the last years of the horizon only build generators with low build costs even if their marginal generation costs are high. This is because the average cost of generation between build year and the end of the planning horizon, including build costs, will be lower for such generators.
The option to control the treatment of the last year is set by the LT Plan End Effects Method attribute. By default we assume that the last year of the horizon is repeated an infinite number of times. The objective function is expanded by the cost of the years after the final horizon year, which equals the sum of the present values of generation costs and annualized build costs. By including annualized build costs in the perpetuity we are assuming that the system continues to expand after the final year.
The following table shows the discount factors applied to each year of a 10-year horizon assuming a discount rate of 12%, and showing the perpetuity applied to the last year.
| Year | Formula | Discount Factor |
|---|---|---|
| 1 | 1/(1+0.12)1 | 0.892857143 |
| 2 | 1/(1+0.12)2 | 0.797193878 |
| 3 | 1/(1+0.12)3 | 0.711780248 |
| 4 | 1/(1+0.12)4 | 0.635518078 |
| 5 | 1/(1+0.12)5 | 0.567426856 |
| 6 | 1/(1+0.12)6 | 0.506631121 |
| 7 | 1/(1+0.12)7 | 0.452349215 |
| 8 | 1/(1+0.12)8 | 0.403883228 |
| 9 | 1/(1+0.12)9 | 0.360610025 |
| 10 | 1/(1+0.12)10 + (1/(1+0.12)10)/0.12 | 3.005083542 |
The formulation of battery (storage) expansion is quite similar to a generator expansion candidate. The charging and discharge power are regulated by existing and expansion units:
Bat_Charge(g,t) ≤ MaxLoad ( Unitsg + ∑ i≤y Bat_Build g,i )The maximum volume (energy capacity) of the battery is also expanded if new units are built and allows battery to store more energy.
EndVolume(g,t) =< Max Capacity*(Unitsg + ∑ i≤y Bat_Build g,i)In addition, at the beginning of the build period (yearly/quarterly/monthly), if a battery unit is built, the initial stored energy (initial SOC) is added to the existing energy level:
EndVolume(g,0) = EndVolume(g,0) + initial SOC*Max Capacity*(∑ i≤y Bat_Build g,i)