### Scheduling of Power Generation: A Large-Scale Mixed-Variable Model

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The thermal reserve margin of each period is the total available thermal capacity divided by the period peak load. The initial and final hydro reserves are specified by peAods by maintenance. This constraint represents the con- Maximum system outage capacity simultaneously in dition imposed to provide some amount of power available maintenance to account for increments in demand or failures of commit- ted generation units.

This high level language allows a fast imposed to the pumped energy. The relaxed linear optimization is carried Generation constraints out through an interior point method called primal-dual For each thermal unit the maximum generation is less predictor-corrector, changing to simplex once optimum for than the maximum available capacity and the minimum a basic solution has been reached.

Crossover option is generation is greater than the minimum load. Thermal unit used for restarting from optimal continuous solution. In this case, those in a lower load subperiod weekends. OPTCR is set to 0. Available in OSL, strategies 1 and 2 are selected. They perform probing only on satisfied variables and using previous integer solution founded.

Preprocessing and probing techniques can be very effec- tive in improving the performance of a mixed integer op- timization [ In effect, the idea is reformulate the prob- lem so as to obtain reductions in optimization time linear and mixed integer , and in the gap between the linear and the integer solution. In this case, setting as variable the unavailability of generation units instead of availability, produces advan- tages in both. The implementation is obvious. Scalation is another important feature with effects on 0 1 Ctsp 5 1.

That means, that variables, its derivatives and constraints should be near 1. For this study, all the hydro units are grouped the goal programming process. They produce as an average criterion chosen. The system demand reaches the peak in the 9th week, and the minimum in the 41st week 1. The profile shows is high in fall and I 1. The annual MS obtained for is presented in Fig. TOC1 is More reliable solutions can be obtained in- 2 creasing p. In this case and due to the dimension of the problem, p can not be reduced.

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Optimization times are very different. The first stage reeds s to reach the optimal point and the second '. Time is expressed in seconds for 0. Reserve margins. The system representation is also relevant be- cause of detailed modeling of maintenance and fuel m - scheduling decisions, thermal units operation and unit E commitment.

It is also new because of its application to a e - real power system not a study case. T1b 1; io 2; io Study year weeks 1 demonstrated by comparing the effects that costs and relia- bility have on each other in thermal plants maintenance Fig. It has been shown that different optimization criteria give different optimal MS solutions, the more an- tagonistic the more different. C c The easy formulation of this goal programming sequen- tial method using GAMS is also relevant and new, as well I 6 as the easy way to call the optimizer and its optimization a options.

Silva, E. Brooke, A. Yamayee, Z. She Systems, vol. Among the subjects of her interest International Business Machines Corp. Kralj, B. From to he was a junior research staff at maintenance scheduling of thermal generating units the Instituto Tecnoldgico para Postgraduados.

## Scheduling of Power Generation: A Large-Scale Mixed-Variable Model

From in power systems. Visiting scholar at Stanford University in Mukerji, R. On the weekend k, auxiliary services coefficient of thermal unit t. The load is m coefficient of the thermal installed capacity modeled by a staircase representation.

## Scheduling of power generation : a large-scale mixed-variable model - Semantic Scholar

The capacity of each thermal unit is divided into two Mi number of periods on scheduled maintenance blocks, being the first one the minimum load block. Heat for the thermal unit t.

This model also allows a piecewise represen- - unit t. A thermal pi, gt maximum and minimum rated capacity of plant consists in units in a physical plant. Fuel con- thermal unit t. For each unit, the number of periods that the mainte- R- power reserve margin. Very small hydro units are aggregated.

Spatial dependen- cies among hydro plants are considered irrelevant to the rl - startup cost of thermal unit t. Therefore, the variation in the hydro energy re- thermal plant c. Only the economic use of pumped units is considered. This economic function includes the transference of energy upper reservoir limit of pumped-storage unit b. Following is the notation and the detailed mathematical performance of pumped-storage unit b. Notation commitment decision of thermal unit t i n uw number of pumped-storage units. D, duration of load level n of subperiod s of S,, fuel storage capacity of thermal plant c at the - period p.

Formulation Contiguity constraints 1,I Objective function of the FIRST stage The MS problem requires contiguity to be modeled This objective function represents the sum of fuel costs when the duration of maintenance works exceeds the time including independent and linear terms of the heat rate and period of the models. In this prob- of fuel stocks plus some penalties for non served power, lem contiguity is modeled using auxiliary constraints and interruptibi1it.

The constraints are formulated in the levels, subperiods and periods of the time scope. This fact is used by the constraint 8 to establish the logical implica- 2 Objective function of the SECOND stage tion between these two variables. This function is the sum of the slack variables of the Other typical maintenance constraints as manpower and constraint 4 , that is to say, the sum of the differences be- material resources limits or sequence constraints tween reserve margins of consecutive periods.

The thermal reserve margin of each period is the total available thermal capacity divided by the period peak load. The initial and final hydro reserves are specified by peAods by maintenance. This constraint represents the con- Maximum system outage capacity simultaneously in dition imposed to provide some amount of power available maintenance to account for increments in demand or failures of commit- ted generation units.

This high level language allows a fast imposed to the pumped energy. The relaxed linear optimization is carried Generation constraints out through an interior point method called primal-dual For each thermal unit the maximum generation is less predictor-corrector, changing to simplex once optimum for than the maximum available capacity and the minimum a basic solution has been reached.

Crossover option is generation is greater than the minimum load. Thermal unit used for restarting from optimal continuous solution. In this case, those in a lower load subperiod weekends.

OPTCR is set to 0. Available in OSL, strategies 1 and 2 are selected. They perform probing only on satisfied variables and using previous integer solution founded. Preprocessing and probing techniques can be very effec- tive in improving the performance of a mixed integer op- timization [ In effect, the idea is reformulate the prob- lem so as to obtain reductions in optimization time linear and mixed integer , and in the gap between the linear and the integer solution. In this case, setting as variable the unavailability of generation units instead of availability, produces advan- tages in both.

The implementation is obvious. Scalation is another important feature with effects on 0 1 Ctsp 5 1. That means, that variables, its derivatives and constraints should be near 1. For this study, all the hydro units are grouped the goal programming process. They produce as an average criterion chosen. The system demand reaches the peak in the 9th week, and the minimum in the 41st week 1. The profile shows is high in fall and I 1.

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The annual MS obtained for is presented in Fig. The general principles are applicable to any culture facility and at any scale of production. The basic yield relationship with light energy input is shown in Figure The use of the term standard cell density requires explanation. In order to compare yields of different species in a culture system, a common factor based on dry weight biomass of harvested algae is applied.

Different alga species vary widely in linear dimensions and in weight per cell, as already seen in Table 1. Knowing the weight per cell, an equivalent number of cells can be calculated for each species to provide a given biomass. For some of the important species this approximates to:. It is the cell density following harvesting and replenishment of the culture volume with new medium relative to light intensity that will largely dictate growth of the culture in the following h period.

Reference to Figure 25 shows that yield is at a maximum at the optimum PHCD when light energy input is not limiting. Above the optimal PHCD, light becomes increasingly limiting due to the self-shading effect of cells at higher culture densities.

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Photosynthesis decreases, therefore, cell division rate and daily yields decrease. Yield is maximal at a particular light intensity and can be increased or decreased by altering the light energy input. Figure The effect of light intensity on yield of Tetraselmis in l internally illuminated culture vessels.