Fast growing load in power systems associated with a large gap between heavy
load and light load periods, generation scheduling and Unit Commitment (UC)
problem has become a crucial issue in operation time horizon and unit commitment
problem has always been an important research challenge in power systems especially
under restructured environment. In a vertically integrated power system, unit
commitment determines when to start-up or shut-down units and how to dispatch
online generators over a given scheduling horizon in order to minimize the operating
costs, satisfying the prevailing constraints such as load balance, system reserve
requirement, ramp rate limits, minimum up/down time limits (Tsung
and Chen, 2007; Afshar et al., 2007;
Yamin et al., 2007; Dieu and Ongsakul, 2008).
Since, the unit commitment is a mixed integer programming, it is very hard to
get an exactly optimal solution and it has been viewed as a very complex optimization
problem and variant methods have been implemented to solve such a complicated
problem either using classical optimization or heuristic as well as hybrid techniques.
Dynamic Programming (DP) is the earliest conventional optimization method that
can be applied to solve the dissimilar size UC problem. The other classical
optimization methods are as follows: Priority List (PL) (Senjyu
et al., 2003) Lagrangian Relaxation (LR), mixed integer programming
(Daneshi et al., 2008) and Branch and Bound (B
and B). The classical optimization techniques, in general, might not be able
to find a solution within a significant computational time for the medium or
large scale UC problem. These limitations have been redounded to introduce the
heuristic optimization methods (Padhy et al., 1997).
With the emergence of metaheuristic and evolutionary algorithm in modern optimization
technique such as: Simulated Annealing (SA), Tabu Search (TS), fuzzy logic,
Genetic Algorithm (GA), Artificial Neural Network (ANN) (Padhy,
2004) and Ant Colony (AC) (Sum-Im and Ongsakul, 2003)
have been used to solve the UC problem. Moreover, in some methods more than
one algorithm has been incorporated together and forms a hybrid technique to
meet the industry requirements. The hybrid methods are also applied to handle
more complicated constraints and are claimed to have a better performance. In
one hand, evolutionary algorithms may seem simple but their solution might be
suboptimal and on the other hand, they might be complicated with more accurate
results (Padhy et al., 1997). The hybrid methods
such as fuzzy dynamic programming and neural network (Daneshi
et al., 2003), genetic-based neural network, Lagrangian relaxation
associated with genetic algorithm (Yamin and Shahidehpour,
2003) and annealing genetic algorithm (Cheng et al.,
2000a) are experienced to tackle to the UC problem.
In this study a new approach considering next hours demand by minimizing the
operating costs considering prohibited zones is presented. The generating units
may have certain ranges where operation is restricted based upon physical limitations
of machine components or instability, e.g., due to steam valve or vibration
in shaft bearings. Therefore, prohibited operating zones as a prominent constraint
must be considered. By including next hours demand at a scheduling horizon the
online units that are not optimal to be turned off kept continuing. On the other
hand, in the proposed formulation a new objective function that comprises start-up
cost is used in order to select the best chromosomes to get better results.
So, at first units with less start up cost are selected and then generation
units with higher start-up cost may have a chance to be turned on in order to
minimize total scheduling horizon costs.
MATERIALS AND METHODS
Unit commitment involves determining generation outputs of all units from an initial hour to satisfy load demands associated with a start-up and shut-down schedule over a time horizon. The objective is to find the optimal schedule such that the total operating costs can be minimized, while satisfying the load demand, spinning reserve requirement as well as other operational constraint.
Objective function: The outage cost as well as fuel cost of generation units should be considered in power system operation as an objective function of a UC problem. The objective function is a function that comprises the fuel costs of generating units, the start-up costs of the committed units and shut-down costs of decommitted units. The start-up cost is presented in two schemes: hot start-up costs and cold start-up costs, while the shut-down cost is assumed to be fixed. Nevertheless the objective function in common form is expressed by Eq. 1.
is power output of unit i at hour t, ui,t is on or off status of
unit i at hour t, SUCi,t and SDCi,t are start-up cost
and shut-down cost of unit i at time t, respectively, N is number of units and
T is unit commitment horizon.
The fuel costs of generating units and the major component of the operating
costs for thermal units are generally given in a quadratic form as it is shown
in Eq. 2. Operating cost coefficients can be given or they
might be estimated using bidding strategies (Badri et al.,
where, ai, bi and ci are fuel cost coefficients for unit i:
The start-up cost is defined as follow:
where, HSCi and CSCi are hot start-up cost and cold start-up
is minimum down-time of unit I,
is duration during which the ith unit is continuously on and CSTi
is cold start time of unit i.
Operational limitation and constraints: The minimization of the objective function is subjected to a number of system and unit constraints such as: power balance, spinning reserve capacity of generating units, prohibited operating zones, minimum up/down time limit as well as spinning reserve requirement. Initial conditions are needed to be considered in scheduling problem:
||Initial condition: Initial conditions of generating
units include the number of hours that a unit consequently has been on-line
or off-line and its generation output at an hour before the scheduling
||Power balance constraint
where, Dt is demand during hour t
are minimum generation and maximum generation of unit i, respectively.
where, SRt is spinning reserve requirement at time t
||Unit ramp-up constraint:
where, RURi is ramp up rate limit of unit I
||Prohibited operating zones and output limit of a generator
||Unit ramp-down constraint:
where, RDRi is ramp down rate limit of unit i:
Prohibited operating zone: Some on-line generating units have their
generation limit, which cannot be exceeded at any time (Saber
et al., 2009). Moreover, a typical thermal unit may have a steam
valve in operation, or a vibration in a shaft bearing, which may result in interference
and discontinue inputBoutput performance-curve sections, called prohibited operating
zones, as shown in Fig. 1.
Therefore, in practical operation, adjusting the generation output of a unit
must avoid all capacity limits and unit operations in prohibited operating zones.
The feasible operating zones of a unit can be described as follows:
are lower and upper bounds of the jth prohibited zone of unit i and PZi
is the number of prohibited zones of unit i.
Minimum up time limit: Minimum number of hours that a unit must be continuously on-line since it has been turned on:
||(a) Main flowchart of proposed Method (UCPOZ) and (b) GA procedure
considering POZ limit
duration during which the ith unit is continuously on.
Minimum down time limit: Minimum number of hours that a unit must be continuously off-line since it has been turned off
duration during which the ith unit is continuously off.
Solution methodology: The optimization technique consists of some steps that is shown in Fig. 2a: Main flowchart of proposed Method (UCPOZ) b: GA procedure considering POZ limit are explained in the following steps. In each step, related constraints are taken into account while finally the objective function associated with all constraints is minimized via Genetic Algorithm (GA).
Call load and units data
Initialization: At this step an initial population is generated according
to Eq. 5, 7, 10 and 11
such that some information for first hour is obtained from initial condition.
In order to have an efficient program the demands of next
hours should be taken into consideration. When a unit is turned off its status
cannot be changed for
hours, while satisfying the next
hours demand excluding this unit should be examined. If it is not satisfied
for any next
hours, scheduling will be referred to the previous hour for re scheduling in
which the later unit should kept online (Pourakbari-Kasmaei
et al., 2008).
Update units data: Here, units' data like the time that a unit continuously has been on/off according to the previous hour's scheduling is updated.
GA procedure: Genetic algorithm is a random and robust search technique
that guides a population towards an optimum using the principles of natural
evolution. This process is facilitated through a fitness evaluation procedure,
which determines the fitness value of each member of the population the so-called
chromosome. Each chromosome contains a number of gens. In this simulation the
chromosome is corresponds to a plant and a gen is corresponds to a unit. The
robustness of GA and its capability across a broad range of problems make GA
as general problem solving techniques in many applications (Swarup
and Yamashiro, 2002). So, in this study according to the complexity of unit
commitment considering prohibited operating zones (UCPOZ), GA is used to solve
this complicated and non-convex optimization problem. The flowchart of the proposed
GA-based solution approach for UCPOZ is shown in Fig. 2b that
includes the following steps:
Initialize the iteration counter as stopping criterion: In this study according to the number of units the number of iterations is set to 80 and at first the iteration counter is set to one.
Economic dispatch: Economic dispatch determines the output of all online units with the objective of minimum total operating costs at a given hour, which is subjected to the power balance constraint Eq. 4 and output limits Eq. 5. For each chromosome of the generated population in step 2 of the main flowchart the ED is applied and the output power of each gens of chromosome is obtained. A lambda iteration method is applied in this study to determine the optimal economic dispatch.
Prohibited zone check: After ED, for each gens of chromosome, the prohibited operating zone check is taken into consideration. If any of gens violated the POZ, the POZ is applied to that gen and ED is repeated for the aforementioned chromosome.
Fitness evaluation: Here, the fitness value of each chromosome should be calculated. In order to accelerate the convergence of the proposed method the fitness function is adopted as follows:
where, A is the big positive number (assumed 1E+4), chr and itr are chromosomes and iteration counter, respectively.
Since, in scheduling problems the objective is to minimize the operating costs, those units with more expensive start-up costs may have no chance to be turned on before they must be, while they may impose less total operating costs. So, in this study a modified cost function with Modified Start Up Cost (MSUC) that is shown by Eq. 13 is used in order to select the best chromosomes for crossover and mutation and then generate new chromosomes to get an optimum scheduling.
At IEEE 10 unit test system the cold start-up cost (CSC) is held twice of hot
start-up cost (HSC) but at this paper Eq. 14 is used, if
changing of HSC to CSC is not as sharp as a step, at this case the change is
||Mating: The mating process consists of three operators:
selection, crossover and mutation (Haupt and Haupt, 2004)
||Modification: After crossover and mutation processes
for achieving feasible chromosomes two following tasks should be handled
||Chromosomes elimination: Infeasible chromosomes that
can not satisfy the SRR constraint will be eliminated as redundant
||Chromosome modification: Since, the number of chromosomes
must be remained constant, chromosomes with the best fitness are replaced
instead of eliminated chromosomes
Stopping criterion: For stopping GA Procedure it is needed to have a criterion, in this study a constant number of iteration has been used.
Best cost selection: Here, the chromosome with the least cost is selected and the output power for all gens is kept as the best answer. All steps are repeated in scheduling time horizon.
RESULTS AND DISCUSSION
The proposed methodology is implemented to a standard IEEE 10-Unit test system while at first the study is only about a commonly unit commitment problem and finally the prohibited operating zones is taken into consideration as a practical and redundant limitation. The POZ that is employed in the study is not an accurate representation. However, there is no great difficulty in making some changes in the formulations developed so that the proposed approach can employ different dispatch representation.
CASE 1: STANDARD IEEE 10-UNIT TEST SYSTEM
The proposed method has been applied to solve a commonly UC problem that so-called 10-unit base system with the given data presented in the Table 5, where, in this case the POZ limitation is not considered. The result of the units output power is given in Table 1 and total cost comparison of several techniques is shown in Table 2.
The load demand of 10 unit base problem is given in Table 6.
CASE 2: IEEE 10-UNIT TEST SYSTEM CONSIDERING POZ
In practice each generator has its generation limit, which cannot be exceeded at any time. Moreover, a typical thermal unit may have a steam valve in operation, or a vibration in a shaft bearing, which may result in interference and discontinue inputBoutput performance-curve sections, called Prohibited Operating Zones (POZ), so it seems be essential to study the POZ as a redundant limitation. As it can be seen from Table 1, at first hour both of units are generated in POZ and this is so difficult to change these generations according to POZ, but using GA is an efficient method for this purpose. The result of the units' output power is given in Table 3.
Table 3 for 24 h time horizon with a total operating cost 564714$. As the PZ is a practical constraint in the UC problem and has not been considered in the previous literatures, so by comparison of UCPOZ cost with the costs in Table 2. Total cost comparison of several techniques, it is clearly seen that there is no main difference between them which present the effectiveness of UCPOZ.
|| Units output power for the 10 unit case
|| Total cost comparison of several techniques
|| Units output power for the 10-unit case considering POZ
|| Abbreviation of UC solution techniques
|| Unit characteristic and cost coefficient of 10-unit base
|| Load demand of 10-unit base problem
The POZ employed in the paper is not an accurate representation which is given
in Table 5. However, there is no great difficulty in making
some changes in the formulations developed so that the proposed approach can
employ different POZ representation.
In this study a reliable and efficient method using heuristic technique for unit commitment as well as scheduling problem has been presented. On the other hand it has been presented a new approach to select best chromosomes via GA, where the objective function in GA has been comprised start-up cost to give a chance to those units that have higher start-up cost and this yields a wide search area. On the other hand, in this paper prohibited operating zones as a practical constraint has been considered. The proposed method has been successfully applied to a standard 10-unit system and a 10-unit system considering POZ and the satisfactory results are compared with the other methods reported in literature. The results also can offer the usefulness of the proposed method which can consider as a practical technique. The results shown that the PM has the following merits in both UC problem and UCPOZ problem: efficient searching ability, robustness in result.