3. A time series model for the aggregate GB wind output circa 2030
4.3. Linear Program for Stochastic Unit Commitment
The (Mixed Integer) Linear Program defining the SUC formulation is described in the sections that follow. The symbols used are defined in the List of Symbols from page 12.
4.3.1. Objective function and load balance constraint
The SUC linear program is: Minimise
∑
n∈N π(n) cLS∆τ(n)PLS(n) +∑
g∈G Cg(n) ! (4.1) subject to the load balance constraint (4.2), and local constraints for the thermal units (4.11)–(4.13) and storage units (4.18)–(4.20), with all decision variables floored at zero. The generation cost Cg(n) is defined in (4.10). The nodal probability π(n) is definedin section 4.4.2. Note that, unlike most formulations in the literature, we express the objective function as a summation across the nodes in the scenario tree rather than a dou- ble summation across the scenarios and timestages. The latter approach requires non- anticipativity constraints that equate the decision variables at timestages that are shared across multiple scenarios; such constraints are not necessary in our version. The differ- ence between the two approaches is merely a matter of computational housekeeping. In the version preesnted here, values for the decision variables are assigned at each node on the scenario tree; in this manner, non-anticipativity constraints are automatically satisfied across multiple scenarios that share common nodes.
The load balance constraint, applied to all n∈ N at the kth timestep, is:
∑
g∈G Pg(n) +∑
s∈S Psdis(n)−Psch(n) +Pwn(n) +PLS(n)−PWC(n) =Pd(k+ ℓ(n)). (4.2)CHAPTER 4. EFFICIENT STOCHASTIC SCHEDULING FOR SIMULATION OF WIND-INTEGRATED POWER SYSTEMS
4.3.2. Slow-starting thermal unit constraints and cost functions
The local constraints pertaining to thermal units with non-zero startup times are set out in this section. The shutdown and startup decision variables, Ngsdand Ngst, are nominally
integer variables, but can be relaxed to continuous ones to reduce run times. All other decision variables are continuous.
Some of the constraints at node n refer to subsets of the ancestors of n. The subsets are defined as follows. If a generator in group g starts generating at node n, then it must have been started up at a node in the set
Ast g(n) =A(n)∩ n n′ ∈ N ∪ P : τ(a(n))−Tgst<τ(n′)≤τ(n)−Tst g o . (4.3) (Note that, because some nodes may represent longer time intervals than others,Ast
g(n)
may contain more than one node.)
If a generator in group g is shut down at node n, it cannot have started generating at any node in the set
Amug (n) =A(n)∩
n
n′ ∈ N ∪ P : τ(n)−Tgmu<τ(n′) <τ(n)
o
. (4.4)
If a generator in group g is started up at node n, it cannot have been shut down at any node in the set
Amo g (n) =A(n)∩ n n′ ∈ N ∪ P : τ(n)−Tgmo<τ(n′) <τ(n) o . (4.5)
Some linear expressions, referred to within the constraints and objective function, are defined below. The number of generators that start generating at node n is equal to the number of generators that were started up Tstpreviously:
Ngsg(n) =
∑
a∈Astg(n)
Ngst(a) (4.6)
The number of generators that are generating at node n is equal to the number of gener- ators that were generating at node n’s parent, plus the number that started generating at node n, less the number that are shut down at node n:
Ngup(n) =Ngup(a(n)) +Ngsg(n)−Ngsd(n). (4.7)
The number of generators that are off at node n is equal to the number of generators that were off at node n’s parent, plus the number that are shut down at node n, less the number that are started up at node n:
CHAPTER 4. EFFICIENT STOCHASTIC SCHEDULING FOR SIMULATION OF WIND-INTEGRATED POWER SYSTEMS
The total power output and operating costs for each generator group can then be written:
Pg(n) =PgmsgNgup(n) +Pgx(n), (4.9) Cg(n) =CstgN sg g (n) +∆τ(n) cnlgN up g (n) +cmgPg(n) . (4.10)
The following constraints are applied to all thermal unit groups g∈ G, for each node
n ∈ N:
The total power in excess of MSG is limited by the number of generating units and the power output range of each unit:
Pgx(n)≤ N up g (n) Pgmax−P msg g . (4.11)
The number of units that are shut down at node n is limited to the total number of units that were generating at node n’s parent, less the number of units that have been generat- ing for less than Tmuhours:
Ngsd(n)≤ N up g (a(n))−
∑
a∈Amu g (n) Ngsg(a). (4.12)The number of units that are started up at node n is limited to the number of units that were off at node n’s parent, less the number of units that have been off for less than Tmo hours: Ngst(n)≤ N off g (a(n))−
∑
a∈Amo g (n) Ngsd(a). (4.13)The total number of units in each group is specified through the values of the com- mitment variables for all past nodes (n ∈ P) at t=0. For example, if all units are up at the start of the simulation we would specify initial values of Ngup= Ngu, N
off
g = Ngsd =Ngst=0
at the past nodes.
In the proposed formulation, the commitment variables Nst and Nsd are not con- strained to be equal at all nodes with a given lookahead time τ. This contrasts with stochastic formulations that are designed to commit the thermal units for the day-ahead schedule [77–81]. With the rolling planning approach, we allow different commitment variables in different scenarios.
Ramp rate limits can also be modelled in the multi-unit group formulation. Since the average output of each unit in the group (in excess of MSG) at node n can be written
Pgx(n)/N up
g (n), one might wish to use the constraint
−∆τ(a)∆Pgrd≤ P x g(n) Ngup(n)− Px g(a) Nupg (a) ≤ ∆τ(a)∆Pgru
where we have abbreviated a(n)to a. However, this contains a non-linear expression, and hence, cannot be included in a Linear Program. If we are modelling ramp constraints, we
CHAPTER 4. EFFICIENT STOCHASTIC SCHEDULING FOR SIMULATION OF WIND-INTEGRATED POWER SYSTEMS
therefore have to make certain assumptions about the output of individual generators within the group.
The ramp constraints are as follows:
Pgx(n)−Pgx(a)≤∆τ(a)∆PgruN up
g (n) (4.14)
Pgx(n)−Pgx(a)≥ −∆τ(a)∆PgrdNgup(a) (4.15)
where we have again used a ≡ a(n) for brevity. Equation (4.14) means that, if some units came up at node n, the power output of any newly online units is limited to Pgmsg+
∆τ(a)∆Pru
g . Equation (4.15) means that, if any units are shut down at node n, they must
each still be generating at least (Pgx(a)/N up
g (a))−∆τ(a)∆Pgrd at node n. If the group
contains just one unit (Ngu =1) then the interaction of Equations (4.15) and (4.11) means
that the unit must be ramped down to Pgmsgbefore being shut down. (Note however that,
since startup and shutdown ramps between zero output and MSG are not modelled, the power output dynamics during startup and shutdown are rather simplified anyway.)
There are some inconsistencies within the ramp constraints. We assume that the individual outputs within a generator group are not sufficiently diverse that the change in output from one timestep to the next is limited by Pmaxor Pmsgfor some generators, and by ∆Pruor ∆Prdfor others. All units in the group are therefore assumed to be generating similar outputs at all times. But because of the ramp constraints, any newly started units may be unable to reach the output of those that are already running, within a single timestep. This is an inevitable result of lumping the generation of many units into one power output.
When using scenario trees with a short horizon, it is necessary to assign a value to thermal units being online at the leaf (terminal) nodes in order to prevent units being shut down prematurely. This is achieved in the WILMAR system using shadow values of relevant constraints [92]. In the rolling planning environment with a 24-hour horizon in the case studies, we found no benefit in assigning online values at leaf nodes and we therefore do not include them in the formulation presented here. Although the quality of the shutdown decisions at the deeper nodes is compromised by the lack of awareness of online values, the here-and-now decisions (which are the only decisions that are imple- mented in a rolling planning context) are not significantly affected in the case studies.
4.3.3. Constraints and cost functions for fast-start and must-run thermal units
In this section we list the constraints and cost functions for thermal units that are mod- elled as must-run (typically nuclear units), or fast-start, i.e. zero startup time, zero startup cost, and zero minimum up- and down-times (typically OCGT units).
Fast-start units could be modelled using the formulation of section 4.3.2, using
Tgst = Cgst = 0, but it is more efficient to model these simpler units using N up g (n) as
CHAPTER 4. EFFICIENT STOCHASTIC SCHEDULING FOR SIMULATION OF WIND-INTEGRATED POWER SYSTEMS
cisions (i.e. we do not define Ngst or Ngsd). For fast-starting units, we therefore simply
constrain Nupg (n)with
Ngup(n)≤ Ngu. (4.16)
Must-run unit modelling is simpler still. In this case, Ngup(n)is not a decision vari-
able, but a constant:
Ngup(n) =Ngu. (4.17)
In each case, the operating costs can be derived from Equations (4.9)–(4.10) with Cst
g =
0.
Ramp rate limits can be imposed as for slow-starting units using Equations (4.14)– (4.15).
4.3.4. Storage unit constraints
The constraints for each storage unit s ∈ S at each node n ∈ N are set out below. All decision variables are continuous.
Emins ≤Es(n)≤Esmax, (4.18)
Psch(n)≤Pscmax(n), (4.19) Psdis(n)≤Psdmax(n) (4.20)
where Es(n)is the state of charge at the end of the time period corresponding to node n:
Es(n) =Es(a(n)) +∆τ(n) ηscPsch(n)− Psdis(n) ηd s . (4.21)
As with online values for thermal units, we found no benefit in assigning a “water” value to energy left in storage units at the leaf nodes in the case studies, and we therefore do not include water values in the formulation.