• fM
r,s Boundary transmission capacity between zones r and s (i.e. maximum power
flow between zones r and s)
Variables
• xrt Available capacity from technology t in zone r
• g0
rtGeneration level (MW) of technology t in zone r in the unconstrained schedule
(recall, from Section 2.1.2 the unconstrained schedule is the schedule which uses all the cheapest technology available to satisfy demand, assuming as much as necessary can be traded between zones)
• fr,s Power flow from zone r to s. Note, fr,scan be negative, which would represent
a positive flow from zone s to r
• g+
rt Volume of accepted offers from technology t in zone r in the constrained
schedule
• g−rt Volume of accepted bids from technology t in zone r in the constrained schedule
Recall from Section 2.1.2 that the bid price is the amount which a generator will pay to produce 1 less MW of capacity. The offer price is the amount which a genera- tor demands to produce an additional MW of capacity. More information about the particular values of these variables is given in Appendix F.3.2.
3.3
Snapshot Constraint Cost Estimation
The objective of the simulator is to estimate mean constraint costs of a given power system over a given period of time. As mentioned in Section 3.1, this is done by breaking the year down into a series of snapshots and estimating the constraint costs in each. This section will detail how costs are calculated for a single snapshot when
the available capacities of all generators across all zones, xrt, and the demands in each
3.3.1
The Unconstrained Schedule
To calculate constraint costs, first an unconstrained schedule must be created. The unconstrained schedule is an initial schedule which seeks to satisfy all demand with the cheapest available capacity whilst acting as if an unlimited amount of generating capacity can be traded between zones, just as in Section 2.1.2.
The cheapest way to satisfy all demand is to schedule generating capacity in order of offer price (lowest first) until the magnitude of scheduled capacity is equal to the sum of demand across all zones. In Section 2.2 this was easy to calculate, as it would consist of all wind and coal capacity available with the remainder of the unconstrained schedule being made up of enough gas to satisfy the remainder of demand.
The unconstrained schedule is still simple to calculate for a general power system.
The unconstrained schedule for technology t in zone r is denoted by g0
r,t. For a fixed
snapshot, τ , generating capacity is scheduled, starting from the lowest offer price, until all demand across all zones would be satisfied, i.e.
X r dy,r,τ = X r,t g0r,t (3.3.1)
The generation scheduled must not be more than the available capacity, i.e.
0 ≤ g0r,t ≤ xr,t (3.3.2)
There is one marginal technology, tM, whose scheduled generation is between 0 and its
availability. The scheduled output for this technology is allocated between all zones in
proportion to the available capacity xr,tM.
Just as in Section 2.2, there will be a curtailment technology with very high bid and offer prices, which means the unconstrained schedule can always be calculated (though part of this schedule may involve curtailing demand).
3.3.2
The Constrained Schedule
The system is split into nRzones, indexed by r. Generating capacity can be transferred
between zones, but only up to the transfer capacity between the two zones, fM
3.3. Snapshot Constraint Cost Estimation 39
transfer capacity of zero would mean two zones aren’t connected (i.e. cannot directly transfer capacity) but may transfer capacity indirectly (through intermediate zones). As mentioned in Section 2.1.2, it will not always be possible to satisfy all demand in the cheapest way possible (i.e. to follow the unconstrained schedule) whilst obeying the limits of transfer capacity between zones. In this case, it is necessary to constrain off (not use) some generating capacity and constrain on (use instead) some alternative capacity so that demand is met without transmitting more between zones than is physically allowed.
The constrained schedule is the cheapest way to satisfy all demand when taking into account the transfer capacities that exist between zones. In Section 2.2 it was feasible to give a blueprint of how to calculate the constrained schedule, and resulting constraint costs, for any possible state of the system. If there was a power flow from Scotland to England resulting from the unconstrained schedule and this flow exceeded the transfer capacity, then gas capacity (or curtailment if no gas was available) would be constrained on and coal capacity would be constrained off. If there was a power flow from England to Scotland resulting from the unconstrained schedule and the power flow exceeded the transfer capacity, then gas capacity would be constrained off and curtailment would be constrained on in its place.
For a more general system (with more than two zones and tens of generating technolo- gies in each zone) it is not feasible to manually calculate the constrained schedule for any possible state of the system. This subsection will give details of the linear program that can be used to calculate the constraint costs of a general system.
Objective Function
The objective of the constrained schedule is to minimise the constraint costs that arise. This is calculated by minimising the sum of the offer prices of generating technol-
ogy constrained on, c+t g+rt, minus the sum of the bid prices for generating technology
constrained off, c−t g−rt, i.e.
minX rt (c+t g+rt − c− t g − rt) (3.3.3)
Variable Bounds
There are several constraints that must be satisfied when minimising Equation 3.3.3. The first are the bounds of the values the variables describing the power system can take: 0 ≤ g+rt ≤ xrt− g0rt (3.3.4) 0 ≤ g−rt ≤ g0 rt (3.3.5) − fMs,r ≤ fs,r ≤ fMs,r (3.3.6) 0 =X r,t (g+r,t− g−r,t) (3.3.7)
Equation 3.3.4 describes the bounds for the volume of offers constrained on, g+rt. We
cannot constrain on negative amounts, nor can we constrain on more than the available
capacity, xrt, minus what is already scheduled, g0rt.
Equation 3.3.5 describes the bounds for the volume of bids constrained off, g−rt. We
cannot constrain off negative amounts, nor can we constrain off more than we have
scheduled, g0
rt.
Equation 3.3.6 describes the bounds of transfer of generating capacity (flow), fs,r, from
zone s to r. A positive value indicates a flow from zone s to r, whereas a negative value
indicates a flow from zone r to s. The flow cannot exceed the boundary capacity, fM
s,r.
Equation 3.3.7 is the demand balance constraint. This equation ensures that the sum of the volume of offers constrained on is equal to the sum of the volume of bids constrained off. As we initially schedule enough energy to satisfy demand, this equation ensures
our final solution satisfies all demand. Again, in each snapshot there will be the
very expensive curtailment technology available, meaning this equation can always be satisfied.
Boundary Flow and Satisfying Demand
In addition to the variable bounds of the previous sub-subsection, nRfurther constraints
are required to ensure demand in each zone of the power system is satisfied. In words,