An implicit reserve constraint unit commitment (IRCUC) model is presented in this paper. Different from the traditional unit commitment (UC) model, the constraint of spinning reserve is not given explicitly but implicitly in a trade-off between the production cost and the outage loss. An analytical method is applied to evaluate the reliability of UC solutions and to estimate the outage loss. The stochastic failures of generating units and uncertainties of load demands are considered while assessing the reliability. The artificial fish swarm algorithm (AFSA) is employed to solve this proposed model. In addition to the regular operation, a mutation operator (MO) is designed to enhance the searching performance of the algorithm. The feasibility of the proposed method is demonstrated from 10 to 100 units system, and the testing results are compared with those obtained by genetic algorithm (GA), particle swarm optimization (PSO), and ant colony optimization (ACO) in terms of total production cost and computational time. The simulation results show that the proposed method is capable of obtaining higher quality solutions.
During the operation of a power system, the balance of active power is often interrupted by some unpredictable factors, such as a random load change, transmission and transformation of power equipment, and failure of the generating unit. In order to keep the balance of power system, a reasonable amount of spinning reserve needs to be set in advance. During the actual power generation dispatch, the system spinning reserve requirements are predetermined by the knowledge of experienced power system operators and introduced as a set of operating constraints which give better performance of the power system [
To deal with the problem of random failure of the unit, the reserve demand was defined as the largest capacity of the installed generating units within the system. Furthermore, some practical systems for scheduling would assume the spinning reserve to be a fixed percentage of power loads. The selecting principle of deterministic reserve neither considered the various random factors of the operating system nor provided the cost of reserve. Therefore, the system state would be subjective to the experience of the dispatcher and in some cases the system would not operate efficiently from economy aspect. As more and more large-scale renewable energy sources are injected into the grid, the uncertainty of the power system will be significantly increased and the current calculation of the system spinning reserve would be more inefficient. Consequently, Anstine et al. [
Based on the above discussion, the spinning reserve constraint is removed from the UC model and an implicit reserve constraint unit commitment (IRCUC) model is proposed in this paper. In addition, the objective function of this model is extended from the minimal production cost of traditional UC model to the minimal sum of production cost and outage loss. That is, if the spinning reserve capacity is scheduled too much in the planning of power generation, then the outage loss would be reduced while the production cost would be increased; on the contrary, if the reserve capacity is scheduled too little, then the production cost would be decreased while the outage loss would be increased.
Same as the traditional UC model, the IRCUC model proposed in this paper is a combinatorial optimization problem with time and nonlinearity, which contains continuous and discrete variables, so it is difficult to get the analytical solution. To solve the UC problem, extensive researches have been carried out by domestic and foreign scholars. The solutions are mainly attributed to the following categories: (1) priority list (PL) algorithm [
In the solving process of IRCUC model, the reliability of generation schedule is evaluated repeatedly to estimate the power loss. Therefore, compared with the traditional UC model, the solving of IRCUC model will become more difficult. In this paper, a novel heuristic called artificial fish swarm algorithm (AFSA) is introduced into this topic for the first time [
The rest of this paper is organized as follows. Section
Under the satisfaction of the various conditions, the basic task of UC model is to seek the planning of power generation and the arranging output with minimum cost. The total production cost
The minimization of the objective function is subject to a number of system and generating unit constraints.
In order to ensure the reliability of power system, the demanded reserves of all time are predetermined by the dispatcher according to the operating experience and also are introduced into the UC model as constraint conditions. Actually, the demanded reserves are related to various factors. Therefore, it is not easy to predetermine the spinning reserve capacity of the power system. In general, the increased value of
It follows from (
The foundation of solving the IRCUC model is to evaluate the reliability of power generation. The reliability index calculated only depends on considering the load uncertainties and the random failures of units.
Two state models are adopted by the units, which include the normal state and the fault state. The IRCUC model in this paper is mainly used for short-term scheduling of the power system (generally less than 24 hours). The possibility of repairing or replacing the faulty units is ignored in such a short time [
In the IRCUC model, the load
Assuming the load power factor as a constant, then the reactive power load
For calculating the The random fluctuation of power load and the random failures of units are assumed to be the independent random events. Random failure of each unit is independent of other units. Considering that the probability of faulty unit is small, the multiple failures of units are ignored in this paper. In other words, there is one unit out of order at most in every moment.
It is supposed that there are
Once the actual load of power system is greater than the available generating capacity, it would lead to part power outage. Therefore, the value of reliability index EENS at time
According to the above analysis, the IRCUC problem belongs to a class of NP-hard problems, as much, is very difficult to solve. Artificial fish swarm algorithm (AFSA), a novel intelligent algorithm, was first proposed in 2002 [
In a water area, fish are most likely distributed around the region where foods are the most abundant. A fish swarm completes its food foraging process by each functioning several simple social behaviors. It is found that there are three most common fish behaviors: (1) searching behavior, that is, fish tend to head towards food; (2) swarming behavior, gregarious fish tend to concentrate towards each other while avoiding overcrowding; (3) following behavior, the behavior of chasing the nearest buddy. Inspired by swarm intelligence, AFSA is an artificial intelligent algorithm based on the simulation of collective behavior of real fish swarms. It simulates the behavior of a single artificial fish (AF) and then constructs a swarm of AF. Each AF will search its own local optimum, pass on information in its self-organized system, and finally achieve the global optimum.
Suppose the searching space is D-dimensional and there are
In the initial state of the algorithm, the variable of trial number should be defined as the trial times of AF searching for food. Then, the following steps describe the fish swarm behaviors.
Searching behavior of AFSA.
Swarming behavior of AFSA.
Following behavior of AFSA.
In the process of AFSA, searching behavior lays the foundation for the AF, swarming behavior enhances the convergence of stability, following behavior ensures the convergence of quickness, and behavior selection guarantees the high efficiency and stability of the algorithm. Through the behavior selection, the AFSA forms an optimization strategy with high efficiency.
The AFSA has the ability to grasp the searching direction and avoid falling into the local optimum. But when some fish move in aimless random or gather around the local optima, the speed of convergence will be slowed down greatly and the searching accuracy is greatly reduced. To avoid premature convergence, an intelligent mutation operator (MO) similarly to genetic algorithm is introduced to enhance the ability escaping from the local optima in this paper. If the state of AF is not improved during the iterations and the AF has entered into a state of partial mining, it needs to be mutated. The specific steps are as follows. Randomly select one of the variables in the position to plus one, and choose the nonnull to minus one. If the value of state is better than that of the current state, update the state of position; otherwise, go back step (1) until the initial number of the mutating operation is satisfied.
By adding the mutation mechanism into the AFSA, it achieves the aim of altering the AF. Through adjusting the swarms, the rate of convergence and the global searching ability of AFSA are both improved. The selection of mutating probability will have a great influence on the performance of the proposed algorithm, which has a positive correlation to the elapsed time. According to the experimental experience, the probability of mutation operator (PMO) selected as 1/(30
In this paper, the binary coded matrix is used to denote the AF swarm. Each individual of AF swarm corresponds to a start-up planning of IRCUC model.
In this matrix, the element 0 refers to the outage state of the units; on the contrary, the element 1 corresponds to the power-on state. The AF in swarm is divided into two categories: the valid AF and the invalid AF. If the available generating capacity of AF is greater than the load forecasting at any time of the start-up planning, the AF is valid, or is otherwise invalid.
The food concentration of AF in current state is one of the important factors of AFSA, and its value determines the behavior selection in the optimization process. In the solving process of IRCUC model, economic dispatch (ED) should be computed for calculating the food concentration. The essence of ED is following the principle of the lowest fuel cost and assigns the power load to the state variables marked as 1 of units in the start-up planning. Considering that the ED calculation is time consuming, this algorithm only takes ED calculation by the valid AF. The invalid AF in the swarm is directly given poor food concentration and made to be sifted out gradually in the evolution process. After ED calculation, the total fuel cost of power generation is counted by the fuel cost function of each unit, which pluses the start-up cost of units to obtain the corresponding power generation cost of AF. It is needed to point out that the AF in accordance with the start-up planning may violate the constraints of the unit minimum uptime and downtime as mentioned in (
In (
When using the PL algorithm to solve the UC problem, the start-up order of each unit is determined by the average full-load cost of units. According to the start-up order, the MO operation should be adopted in this paper. The detailed descriptions are as follows.
Compute the percentage of the value of reliability index
If it is undersaturation at that time, the outage units should take MO operation (start-up) according to the order of start-up, until the state of that time is not under-saturation.
If it is over-saturation at that time, the operational units should take MO operation (closedown) according to the order of start-up, until the state of that time is not over-saturation. It is noticeable that the MO operation should be tried to avoid violating the constraints of the unit minimum uptime and downtime in (
Note that the MO operation is executed in accordance with a certain probability which was mentioned in Section
Flow chart of solving the IRCUC model with improved AFSA.
The efficiency of the proposed method is verified by solving the IRCUC model. The improved AFSA method is initially tested by systems with different sizes based on a basic system of 10 units from the literature [
For assessing the reliability of the generation schedule, the reliability parameters of units are given in Table
Reliability parameters of units.
Number of unit | Failure rate (times/hour) |
---|---|
1-2 | 0.00033 |
3–5 | 0.00050 |
6–7 | 0.00151 |
8–10 | 0.00075 |
This paper uses the improved AFSA to schedule the planning of power generation and the output arrangement. The parameters of the proposed algorithm are shown in Table
Parameters of the proposed algorithm.
Parameter | Value |
---|---|
Sizepop | 500 |
Step | 0.3 |
Visual | 1.8 |
PMO | 0.03~0.1 |
Try_number | 10 |
MAXGEN | 100 |
Penalty coefficient |
106 |
Upper limit of |
0.5% |
Lower limit of |
0.1% |
|
0.618 |
The solid line in Figure
Total cost in the process of iterations.
For comparison, this paper also uses the traditional UC model to optimize the planning of power generation and the scheduling of output power. In the process of optimization, the spinning reserve is assumed to be 5% and 10% of the load demand in each system. The results are illustrated in Figures
Optimal scheduling of power generation (5%, 10%, and 0% of the load demand, resp.).
Unit no. | Hour | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
5 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| ||||||||||||||||||||||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
4 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 |
5 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| ||||||||||||||||||||||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
5 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Reliability index of optimal generation schedule.
Production cost and outage loss.
The assessments of reliability under the optimal schedule of power generation are provided with each of the scheduling times in Figure
Moreover, it should be emphasized that the IRCUC model is not needed to specify the capacity of spinning reserve in advance. Therefore, the IRCUC model is more practical and flexible than the normal UC model.
In the model of IRCUC, the value of
Scheduling of power generation with different
Unit no. | Hour | |||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
5 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| ||||||||||||||||||||||||
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
3 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
4 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 |
5 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
Production cost and EENS cost with different
It can be seen from Figure
To verify the feasibility and effectiveness of the proposed method for solving large-scale IRCUC model, the simulations are tested on systems with 20, 40, 60, 80, and 100 units, respectively. For the systems with 20, 40, 60, 80, and 100 units, the basic 10 units system is duplicated and the total production costs are adjusted proportionally to the system sizes. To avoid any hazardous interpretation of optimization results, related to the choice of particular initial states, 10 trials for each units system are performed. The best and worst total production costs of the improved AFSA (IAFSA) are obtained with different
Comparison of the total production costs ($) for up to 100 units using IRCUC model with other methods (
Units | GA | PSO | ACO | IAFSA | ||||
---|---|---|---|---|---|---|---|---|
Best | Worst | Best | Worst | Best | Worst | Best | Worst | |
10 | 554,943 | 556,121 | 555,404 | 556,231 | 556,686 | 558,006 | 554,105 | 554,579 |
20 | 1,110,068 | 1,125,790 | 1,119,244 | 1,133,793 | 1,112,292 | 1,117,054 | 1,108,979 | 1,110,643 |
40 | 2,219,103 | 2,229,024 | 2,219,523 | 2,240,085 | 2,220,116 | 2,239,146 | 2,218,163 | 2,223,117 |
60 | 3,340,066 | 3,352,886 | 3,342,108 | 3,359,012 | 3,340,407 | 3,353,921 | 3,338,979 | 3,343,125 |
80 | 4,460,202 | 4,472,478 | 4,465,349 | 4,480,939 | 4,465,177 | 4,477,753 | 4,457,032 | 4,461,943 |
100 | 5,583,977 | 5,658,925 | 5,617,388 | 5,680,841 | 5,590,607 | 5,651,827 | 5,581,284 | 5,608,291 |
Comparison of the total production costs ($) for up to 100 units using IRCUC model with other methods (
Units | GA | PSO | ACO | IAFSA | ||||
---|---|---|---|---|---|---|---|---|
Best | Worst | Best | Worst | Best | Worst | Best | Worst | |
10 | 537,582 | 540,312 | 537,515 | 549,530 | 538,797 | 546,660 | 536,837 | 537,279 |
20 | 1,076,156 | 1,085,332 | 1,088,494 | 1,102,101 | 1,078,919 | 1,090,458 | 1,075,279 | 1,079,034 |
40 | 2,152,751 | 2,168,248 | 2,159,903 | 2,180,850 | 2,160,012 | 2,174,762 | 2,148,968 | 2,155,008 |
60 | 3,238,565 | 3,259,686 | 3,240,161 | 3,278,542 | 3,245,471 | 3,269,129 | 3,232,139 | 3,240,654 |
80 | 4,388,004 | 4,426,215 | 4,395,974 | 4,438,793 | 4,395,848 | 4,434,752 | 4,381,521 | 4,388,022 |
100 | 5,428,795 | 5,479,510 | 5,439,588 | 5,492,534 | 5,430,955 | 5,485,750 | 5,407,844 | 5,429,125 |
As shown in Tables
In addition, the computational times of the improved AFSA and other methods are another important evaluation index. The computational times of the above methods to find the optimal solutions with various numbers of units to be committed are shown in Table
Comparison of the computational times (s) for up to 100 units using IRCUC model with other methods.
Units | Computational times | |||
---|---|---|---|---|
GA | PSO | ACO | IAFSA | |
10 | 90 | 55 | 25 | 19 |
20 | 255 | 158 | 60 | 45 |
40 | 728 | 440 | 180 | 95 |
60 | 995 | 885 | 355 | 182 |
80 | 1,780 | 1,624 | 595 | 242 |
100 | 3,310 | 2,825 | 860 | 448 |
It follows from Table
To overcome the deficiency of UC model, this paper tried to remove the constraint of spinning reserve from the traditional UC model and the IRCUC model was proposed. The objective function was extended from the minimal production cost of traditional UC model to the minimal sum of the production cost and the outage loss. The demand and reserve constraints were not explicitly given in the IRCUC model. Rather, the tradeoff between the production cost and the outage loss of the objective function was implicit.
The IRCUC model was solved by using the AFSA. In addition to the standard operation of AFSA, the MO operation was designed with the idea of PL algorithm and dramatically improved the optimizing ability of AFSA.
The efficiency of the proposed method was demonstrated by using the testing systems with the number of generating units from 10 to 100. The numerical results showed the improvements in effectiveness and computational time compared to the results obtained from other methods. Furthermore, the results proved that the method is capable of solving realistic UC problems.
This work is supported by the Graduate Education Innovation Project in Jiangsu Province (no. CXZZ12_0228). The authors would like to thank the editor and anonymous reviewers for their suggestions in improving the quality of the paper.