Results

Let us now take a look at the results generated by the three models. The Original Unit Commitment Problem is used as the baseline scenario and the results of the two robust versions are compared against it.

For all three model formulations, a feasible unit commitment which satisfies the technical constraints and supplies the load is found.

The characteristics of the power generation units used in the test instances (number of units per type, minimum and maximum generation levels, fixed cost when running at minimum level, variable cost between minimum and maximum generation levels, start-up cost) are summarized in Table Description of power generators.

Table Nominal case presents results of the nominal optimization problem. For each time period, the number of started units is presented (Up Units), along with the generation power (Gen. Pwr.). The generation capacity (Gen. Cap.) is the total maximum power generation available from the started units. The columns 'reserve down' (Res. Dn.) and 'reserve up' (Res. Up) respectively show the maximum load decrease or load increase that can be safely supported without requiring startup or shutdown of any units.

For example, during the first period (0-6) the load can increase from 12000 kWh (power demand forecast) to 13250 kWh (power demand forecast + reserve up) without having to start up new units.

The last two lines present the maximum loss-of-load in case the worst scenario realizes. For the demand variation, the worst scenario is the realization of the load profile that requires the highest power generation increase. For the contingency planning, the worst scenario occurs when the k 'critical' units are forced in outage. The criticality of a unit depends on its power generation level and total reserve. A unit without reserve up is critical, as is a unit supplying a very large share of the total load.

If the nominal case is applied, then in case of demand variation, there is a risk of not supplying 1395 kWh of demand during the morning peak. Common sense would be to start up a new unit in order to cover the risk for all time periods when there is a risk of loss of load. However, the results from the robust formulation for the demand variation problem presented in Table Robust against demand variation show that it is possible to overcome the risk merely by changing the type of committed units.

The loss of load for the 2 contingencies case highlights that there is a risk of not being able to supply 2750 kWh during the initial periods. It means that about 20% of the total demand will not be covered. Indeed there is only 1250 kWh of upward reserve available, which means that even if just a single unit running at full capacity is lost (1500 kWh) then the power demand cannot be supplied. Like in the previous case, the common sense conclusion would be to start up new units. However, the results from the robust formulation for the 2 contingencies problem presented in Table Robust against 2 contingencies show that it is possible to find a unit commitment schedule without risking load disconnection and without committing too many new units (and hence incurring higher startup costs).