Optimizing a unit commitment problem using an evolutionary algorithm and a plurality of priority lists

Abstract

The Unit Commitment Problem (UCP) is an operational research problem commonly encountered in energy management. It refers to the optimum scheduling of the generating units in a power system to efficiently meet the electricity demand. UCP comprises two interrelated sub-problems: the Unit Commitment for deciding the operating state of the units at each scheduling period and the Economic Dispatch (ED) for allocating the demand among them. Various Evolutionary Algorithms (EA) have been adopted for solving UCP, commonly assisted by the Lambda iteration method for solving the ED. In this study, an EA-based method is proposed for dealing with both sub-problems, avoiding binary variables through a simple transformation function. The method takes advantage of a repair mechanism utilizing the Priority List (PL) to steer the search towards adequate generating schedules. The impact of the cost metric chosen for creating the PL on the computational results is investigated and the use of a Plurality of PL is suggested to alleviate the biases introduced by employing constant cost metrics. Furthermore, an Elitist Mutation strategy is developed to enhance the performance of the proposed EA-based method. Simulation results on various power systems validate the beneficial effect of the proposed modifications. Compared to state of the art, the algorithm proposed has been at least equivalent, exhibiting consistently solutions of lower or competitive costs in all systems examined.

Appendix

1.1 Appendix 1: List of symbols

The symbols used in this study are presented in the following list:

$a_{i}, b_{i}, c_{i}$:

Fuel cost function coefficients of unit i, ($\$/h$), ($\$/MWh$) and ($\$/MWh^{2}$)

A:

The external archive used in FROFI

$AVC_{n,i}^{g}$:

Average production cost calculated for unit i of particle n at generation g , ($\$/MWh$)

CR:

Crossover rate

$CSTC_{i}$:

Cold start-up cost of unit i, ($)

F:

Scaling factor of mutation

$FC_{i}$:

Fuel cost of unit i, ($)

$G_{i}^{t}$:

Power generated by the ith unit at tth time period,(MW)

$Gop_{n,i}^{g}$:

Operating point for calculating the average cost of the ith unit of particle n at generation g, (MW)

i:

Index of thermal generating units

g:

Generation index in Differential Evolution

$Gmax_{i}$:

Maximum power limit of unit i,(MW)

$Gmean_{i}$:

Mean operating point of unit i,(MW)

$Gmin_{i}$:

Minimum power limit of unit i,(MW)

$HSTC_{i}$:

Hot start-up cost of unit i, ($)

$MDT_{i}$:

Minimum up time of unit i, (h)

MRN:

Maximum number of replacements in the Replacement Mechanism

$MUT_{i}$:

Minimum down time of unit i, (h)

n:

Index of population member

N:

Number of generating units

NPop:

Size of the population in DE

$P_{D}^{t}$:

Expected load demanded at period t, (MW)

$P_{R}^{t}$:

System spinning reserve at hour t, (MW)

$RD_{i}$:

Ramp down ability of unit i, (MW / h)

$RGmax_{i}^{t}$:

Restricted maximum power limit of unit i at hour t due to its ramping capabilities, (MW)

$RGmin_{i}^{t}$:

Restricted minimum power limit of unit i at hour t due to its ramping capabilities, (MW)

$RU_{i}$:

Ramp up ability of unit i, (MW / h)

$SDC_{i}^{t}$:

Cost of shutting down unit i at time period t, ($)

$STC_{i}^{t}$:

Cost of starting up unit i unit at time period t, ($)

t:

Index of time periods

T:

Number of planning periods, (h)

$Tcold_{i}$:

Time interval needed for unit i to cool down, (h)

$Toff_{i}^{t}$:

Time-span, during which unit i is continuously off up to period t, (h)

$Ton_{i}^{t}$:

Time-span, during which unit i is continuously on up to period t, (h)

$u_{n}^{g}$:

nth trial vector at generation g

$UL_{n}^{g}$:

Percentage defining the operating point of the units used to create the Priority List for particle n at generation g

$v_{n}^{g}$:

nth mutant vector at generation g

$x_{n}^{g}$:

nth target vector at generation g

$Y_{i}^{t}$:

Operating status of ith unit at tth time period (on-line = 1, off-line = 0)

$\chi _{i}, \delta _{i}, \gamma _{i}$:

Cost coefficients of the exponential start-up cost function of unit i

1.2 Appendix 2: Description of the replacement mechanism of FROFI

In the replacement mechanism some individuals of the current population will be replaced by the individuals in A. The individuals are sorted based on their objective function value, and the sorted population is divided into MRN parts of equal size. The individual with the maximum total constraints violation, $\vec {x}_{a}$, in the first part and the individual with the minimum total constraints violation in A, $\vec {x}_{b}$, are selected. If $f(\vec {x}_{b}) < f(\vec {x}_{a})$, then $\vec {x}_{b}$ replaces $\vec {x}_{a}$ in the current population and is deleted from A. The same procedure is carried out for the second part of the sorted population and the individual with the minimum total constraints violation in A. The replacement mechanism is terminated when either all the MRN parts have been updated or A has become an empty set.

1.3 Appendix 3: Data of the system of 10–100 units

The basic system of 10 units was introduced in Kazarlis et al. (1996). The specifications of the units are presented in Table 14. It is noted that the initial state of the units is taken into consideration. In the last column of Table 14 a positive (negative) integer represents the number of hours for which a unit has been on (off) prior to the first hour of the scheduling period. The daily load demand for the same system is given in Table 15.

Table 14 Unit characteristics for the system of 10 units

Table 15 Hourly load demand for the system of 10 units