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Mathematical formulation

The set Horizon of consecutive time periods describes the planning horizon of the study. Time periods t∈Horizon have different lengths LENt. Each time period t∈Horizon is associated with a power demand DEMt.

The set Units denotes the set of available power generation units. The minimum power generation level of a unit is PMINu, and its maximum capacity is PMAXu. The startup cost of a single unit is CSTRu, and the cost of running it at the minimum power generation level is CMINu. The marginal power production cost above PMINu for each hour is CADDu.

Each unit u is associated with three decision variables. The binary variable startut equals 1 if the unit u is starting at the beginning of time period t. The binary variable workut equals 1 if the unit u is up and running during the whole time period t. At last, the variable paddut is set to the power generation level of the unit u during time period t.

Original Unit Commitment Problem

The objective function, to be minimized, is the expected operation cost. The operation cost is composed of the startup cost and the generation cost. It can be expressed as follows:

∑
u∈Units,t∈Horizon
CSTRu ·startut + LENt·(cminuworkut + caddu·paddut)

The startup constraints describe the time period(s) during which the unit is starting. They can expressed as follows:

startut ≥ workut - workut-1,  ∀u∈Units, if   t>1
startu1 ≥ worku1 - workuT,  ∀u∈Units, if   t=0

The maximum power generation level constraints limit the power generation level of a unit. These constraints are stated as:

paddut ≤ (PMAXu - PMINu)·workut  ∀u∈Units, t∈Horizon

The power balance constraints ensure that the total power production equals the power demand at any time. They are formulated by these equations:

∑
u∈Units
PMINu·workut + paddut = DEMu,  ∀t∈Horizon

Load Robust Unit Commitment Problem

The Load Robust Unit Commitment problem extends the Original Unit Commitment problem by constraining the unit commitment to be safe under power demand variation. Historical power demands for the last years are known.

The scenario uncertainty set
Let Udem be the set of possible future power demands. Let Years be the set of past years that should be taken into account, and HDEMyt the demand for the time period t of the year y. Then the uncertainty set can be expressed as follows:

Udem = { e : ∀t∈Horizon, ∃y∈Years : et = HDEMyt }

Robust constraints

∑
u∈Units
PMAXu · workut ≥ demt  ∀dem∈Udem, t∈Horizon

The reformulation engine will efficiently handle the resulting, potentially large number of constraints, even with large sets of historical data.

The N-k Contingency-Constrained Unit Commitment Problem

The N-k Contingency-Constrained Unit Commitment problem extends the Original Unit Commitment problem as to make sure that the committed power generation units can supply the load, even if k generators are simultaneously forced in outage. The uncertain eut represents the forced outage event, and the uncertainty set is the set of units that are in forced outage.

The power generation units cannot efficiently be turned on or off at short notice. At the time of scheduling the units, the demand is not known for sure and we must make sure that the total maximum power output capacity of the committed units is greater than the highest demand forecast. Similarly, the total minimum power generation level must be lower than the lowest demand forecast.

The cardinality uncertainty set
Let Uoutage be the set describing groups of combinations of k units in forced outage. This uncertainty set can be expressed as follows:

Uoutage = { e :
∑
u∈Units
eu ≤ k }

Robust constraints

∑
u∈Units
PMAXu * workut · (1-eu) ≥ DEMt  ∀e∈Uoutage, t∈Horizon
∑
u∈Units
PMINu * workut · (1-eu) ≤ DEMt  ∀e∈Uoutage, t∈Horizon

These two robust constraints contain the same uncertains. Because uncertains are used to model worst case situation for a particular constraint, the same uncertain may have different values in each constraint.

Parent Topic Robust unit commitment

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