In this section, we present verbally and formulate mathematically the affine objective function and the constraints for the UC optimization problem. The discrete part of the problem will be formulated with three binary variables. Thus, we formulate a binary mixed integer linear programming problem.
Consider a set I :={1,2, . . . ,I} of power generating plants and a specified time-varying demand overT :={1,2, . . . ,T} time periods defining the planning time hori-zon. The mathematical programming model involves I×T continuous nonnegative real variables,pi,t ∈R≥0; and three I×T binary variables: ui,t,yi,t,zi,t∈Z2.
4.2.1 Objective function
The objective cost function is formulated to minimize the total operating power pro-duction cost. The operating cost consists of running cost, startup cost, and shutdown cost.
The running cost is model by fixed and variable cost. The fixed cost is expressed as
aiui,t, (4.1)
whereai is the fixed cost of plantiand ui,t is a binary variable that is equal to one if plantiis committed during time periodtand zero otherwise. The variable cost is expressed as proportional to the plant power output:
bipi,t, (4.2)
wherebi is the variable cost of plantiandpi,t is the nonnegative real variable that is the power output of plantiduring time periodi.
The startup and shutdown cost is considered constant. Every time a plant is started up, its startup cost is added. Similar, every time a plant is shut down, its shutdown cost is added. Thus, we obtain
SUiyi,t+SDizi,t, (4.3)
whereSUi andSDi are the startup and shutdown cost of planti, respectively. yi,t is a binary variable that is equal to one if plantiis started up at the beginning of time periodi and zero otherwise andzi,t is a binary variable that is equal to one if plant iis shut down at the beginning of time periodiand zero otherwise.
The function to be minimized is obtained by combining (4.1)–(4.3), thus, the objective function of the UC problem is
ϕ=∑
i∈I
∑
t∈T
[aiui,t+bipi,t+SUiyi,t+SDizi,t]. (4.4)
4.2 Mathematical problem formulation 23
4.2.2 Constraints
Several constraints may be placed to the UC problem, as many different requirements can be given; e.g., individual requirements on power demand, reliability, physical limits of equipment, power system operating limits, etc. The list presented here is by no means exhaustive; these are the ones being considered and implement in this thesis.
To deduce some of the constraints, we use logical equivalences; seeAppendix A.2.1.
4.2.2.1 Power demand load balance
The system power demand should be satisfied for each time period:
∑
i∈I
pi,t≥Dt, t∈ T, (4.5)
where Dt is the power demand in time period t. Power contribution from non-controllable renewable power sources will change the need for producing power from the conventional power plants. P Wt represent forecasted power production from renewable power sources at time periodt. Thus,(4.5)i extended to
∑
i∈I
pi,t≥Dt−P Wt, t∈ T. (4.6)
4.2.2.2 Spinning reserve
In order to ensure reliability in term of enough resources available during the real-time operation of the power system, the system operator allocates reserve capacity to cover unexpected shortages of energy supply in real-time.
The required spinning reserve should be guaranteed to be available by the
com-mitted plants: ∑
i∈I
P Uiui,t ≥Dt+Rt, t∈ T. (4.7) whereP Ui is the maximum power output generation of plantiandRtis the required spinning reserve at time periodt. Like the demand load balance constraint above, we introduceP Wtin the event of contribution from renewable power sources, thus,
∑
i∈I
P Uiui,t≥Dt+Rt−P Wt, t∈ T. (4.8)
4.2.2.3 Power output limitations
The power plants are limited within an operating range, i.e., if a plant is committed, the power output is to be within its minimum and maximum power output generation.
This may be expressed as
P Liui,t≤pi,t ≤P Uiui,t, i∈ I, t∈ T, (4.9)
24 4 Unit Commitment
whereP LiandP Uiare the minimum and maximum power output generation of plant i, respectively. We see, if planti at time periodt is committed,ui,t = 1, the power output,pi,t, is to be within limits, whereas if plantiat time periodtis decommitted, ui,t= 0, the preceding constraint forcespi,t= 0.
4.2.2.4 Ramping rate limitations
The power plants ability to increase and decrease to higher and lower power output from time periodktok+1is limited. The so called ramp rate limits may be expressed by
pi,t−pi,t−1≤RUi, i∈ I, t∈ T, (4.10) pi,t−1−pi,t ≤RDi, i∈ I, t∈ T. (4.11) For the time periodt = 1, pi,0 is given by the initial output power of planti. RUi
andRDi are the maximum ramp-up and ramp-down limit of planti, respectively.
4.2.2.5 Startup and shutdown
Any committed plants can be shut down but and not started up, and analogously, any decommitted plants can be started up but not shut down. This can be expressed by logic constraints with startup and shutdown cost term added, respectively:
ui,t−ui,t−1≤yi,t, i∈ I, t∈ T, (4.12) ui,t−1−ui,t≤zi,t, i∈ I, t∈ T. (4.13) For the time periodt= 1,ui,0is given by the plants status preceding the first period of the planning horizon. By considering the possible scenarios, the logic constraints (4.12) and (4.13) gives intuitively sense. E.g., consider the lefthand side of (4.12).
The only scenario this yields to one is when ui,t = 1 and ui,t−1 = 0, thus, startup cost should be added to the objective function. The expressions, however, may be derived from logic conditions [RG91]. Consider the startup scenario. Startup cost should be added to the objective function ifui,t = 1and ui,t−1= 0. LetPA denote a committed plantiat timet,¬PB denote a decommitted plantiat timet−1, and PC =yi,t denote whether startup cost is add, yi,t = 1, or not, yi,t = 0. Then, we have
PA∧ ¬PB ⇒ PC By(A.11), we can remove the implication, thus
¬(PA∧ ¬PB)∨PC. By applying(A.12)(De Morgan’s theorem), we have
¬PA∨PB∨PC.
4.2 Mathematical problem formulation 25
With the implication from above and that, e.g.,¬PA= 1−ui,t, the conjunction form can be translated into its equivalent mathematical linear form:
1−ui,t+ui,t−1+yi,t ≥1 ⇒ ui,t−ui,t−1≤1,
which is equivalent to(4.12). Likewise,(4.13)can be deduced by same approach.
4.2.2.6 Minimum up- and downtime
Due to physical characteristics, power plants may not immediately be able to startup and shutdown and vice versa. LetT Ui denote the minimum uptime for planti, once it has started up. Ifyi,t= 1, thenui,t+1= 1,ui,t+2= 1,. . .,ui,t+T Ui; thus, we write
Translating(4.15)conjunction expression into its equivalent mathematical linear form gives the minimum uptime constraint:
1−yi,t+ui,t+j ≥1 ⇒
ui,t+j ≥yi,t, j∈ Ui. (4.16)
Similar, we derive the minimum downtime. LetT Di denote the minimum downtime for plant i, once it has been shutdown. If zi,t = 1, thenui,t+1 = 0, ui,t+2 = 0, . . ., ui,t+T Di, which leads to the logic expression
zi,t ⇒ ∧
4.2.2.7 Restricting carbon dioxide emission
There may be restricting on carbon dioxide emission when generating power. This
may be expressed as ∑
i∈I
∑
t∈T
ECipi,t≤EU, (4.19)
where ECi is the CO2 emission rate for planti andEU denote the maximum CO2
emission allowed.
26 4 Unit Commitment